Apeirology Wiki apeirologywiki https://apeirology.com/wiki/Main_Page MediaWiki 1.43.0 first-letter Media Special Talk User User talk Apeirology Wiki Apeirology Wiki talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk Module Module talk 0 1 1 2022-08-31T13:51:58Z MediaWiki default 1 Create main page wikitext text/x-wiki __NOTOC__ == Welcome to {{SITENAME}}! == This Main Page was created automatically and it seems it hasn't been replaced yet. === For the bureaucrat(s) of this wiki === Hello, and welcome to your new wiki! Thank you for choosing Miraheze for the hosting of your wiki, we hope you will enjoy our hosting. You can immediately start working on your wiki or whenever you want. Need help? No problem! We will help you with your wiki as needed. 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The bureaucrat(s) might still be working on a Main Page, so please check again later! 21236ac3f8d65e5563b6da6b70815ca6bf1e6616 2 1 2022-08-31T15:08:37Z Augigogigi 2 wikitext text/x-wiki <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite. <!-- Quote Div --> <div style="text-align:center;"> <big><big><big>''{{Random_Quote}}''</big></big></big> </div> <br><br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Wiki Links === </div> * [[List of Functions]] * [[List of Ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[http://cantorsattic.info/Cantor%27s_Attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> 1aea875bc6e6bd5f3af85a23e08621f7cac78d53 3 2 2022-08-31T15:10:08Z Augigogigi 2 wikitext text/x-wiki <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Wiki Links === </div> * [[List of Functions]] * [[List of Ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[http://cantorsattic.info/Cantor%27s_Attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> c9f152c341a220cee6319e0c4490a2a562520cc6 30 3 2022-09-05T20:32:05Z Metachirality 7 wikitext text/x-wiki <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[http://cantorsattic.info/Cantor%27s_Attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> 7c9bcb08593c9f92c093abf376b7481197f9d1bd 2 2 4 2022-08-31T15:27:46Z Augigogigi 2 Created page with "Hello! I'm Augigogigi, you may know me as TGR or Augi." wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. 926cd2b8582290cf8ccf222d6ff3d75fde095f12 5 4 2022-08-31T15:36:57Z Augigogigi 2 wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. <span style="color:red;">'''''<big><big><big>BUY SBOCF CHEAP ONLY $1799.99</big></big></big>'''''</span> c6d6cb092154fef247090383c2e9aa7d1bfe8e33 26 5 2022-09-05T19:15:04Z Augigogigi 2 wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. <span style="color:red;">'''''<big><big><big>BUY [[User:Augigogigi/SbOCF|SBOCF]] CHEAP ONLY $1799.99</big></big></big>'''''</span> [[User:Augigogigi/TAN|TAN]] [[User:Augigogigi/TGRx|TGRx]] 6b2dffd76f4108b6c3d1b065a15ccde0b113928a 28 26 2022-09-05T20:22:09Z Augigogigi 2 wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. <span style="color:red;">'''''<big><big><big>BUY SBOCF CHEAP ONLY $1799.99</big></big></big>'''''</span> == My Stuff == [[User:Augigogigi/TAN|TAN]] [[User:Augigogigi/TGRx|TGRx]] [[User:Augigogigi/Mahlo_Notation|Mahlo Notation]] [[User:Augigogigi/SbOCF|SbOCF]] 67071b2cfb2e0aa7e6a29d7fb7eb2d15f1d27d6a 6 3 6 2022-08-31T15:49:41Z Augigogigi 2 wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 0 4 7 2022-08-31T21:30:37Z Augigogigi 2 Redirected page to [[List of ordinals]] wikitext text/x-wiki #REDIRECT [[List_of_ordinals]] db3e4b723a2080b3f02d7a252483807803233f83 8 6 11 2022-08-31T21:52:10Z Augigogigi 2 tex! javascript text/javascript $(function () { mw.loader.load('https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js', 'text/javascript'); }); MathJax = { tex: { inlineMath: [ // start/end delimiter pairs for in-line math ['\\(', '\\)'] ], displayMath: [ // start/end delimiter pairs for display math ['\\[', '\\]'] ] } }; 8fadcdc9b0447c1ed213e7757decf8083a82bd9d 0 9 17 2022-08-31T22:17:46Z Augigogigi 2 testing DISPLAYTITLE wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} '''\(\omega\)''' (also "omega") is the first transfinite ordinal, first limit ordinal, and first admissible ordinal. dd9ec3d1e913f4c2b54b00c4292b40d6754c8a63 53 17 2022-09-06T01:59:11Z OfficialURL 10 Added some more info wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} '''\(\omega\)''' (also "omega") is the first [[infinite ordinal,]] first [[limit ordinal,]] and first [[admissible ordinal.]] It is the [[order type]] of the natural numbers <math>\mathbb N</math>. As a [[von Neumann ordinal]], it corresponds to the naturals themselves. 54fa26cf4b8b9aa50317434a95bd420908f0b4dd 54 53 2022-09-06T02:01:48Z OfficialURL 10 wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} '''\(\omega\)''' (also "omega") is the first [[infinite ordinal]], first [[limit ordinal]], and first [[admissible ordinal]]. It is the [[order type]] of the natural numbers <math>\mathbb N</math>. As a [[von Neumann ordinal]], it corresponds to the naturals themselves. 4ad2570b8b8d6d9a27c6f9377847c4b6c34ccf34 55 54 2022-09-06T02:09:22Z OfficialURL 10 Added properties wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. e423dee0110b04a7f0fb5be58e104b9e861ebe2f 56 55 2022-09-06T02:11:13Z OfficialURL 10 wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. efe9523840585020d3cecfe546cc4de617a99d1b 0 11 21 2022-09-01T14:02:00Z Augigogigi 2 Created page with "The Epsilon numbers are (todo) == <span id="zero">\(\varepsilon_{0}\)</span> == == <span id="one">\(\varepsilon_{1}\)</span> == == <span id="omega">\(\varepsilon_{\omega}\)</span> == == <span id="epsilonzero">\(\varepsilon_{\varepsilon_{0}}\)</span> ==" wikitext text/x-wiki The Epsilon numbers are (todo) == <span id="zero">\(\varepsilon_{0}\)</span> == == <span id="one">\(\varepsilon_{1}\)</span> == == <span id="omega">\(\varepsilon_{\omega}\)</span> == == <span id="epsilonzero">\(\varepsilon_{\varepsilon_{0}}\)</span> == 63323d78f799b1b236b9ece98855a822ff0a0929 22 21 2022-09-01T14:05:07Z Augigogigi 2 wikitext text/x-wiki The Epsilon numbers are (todo) == <span id="zero">\(\varepsilon_{0}\)</span> == == <span id="one">\(\varepsilon_{1}\)</span> == == <span id="omega">\(\varepsilon_{\omega}\)</span> == == <span id="epsilonzero">\(\varepsilon_{\varepsilon_{0}}\)</span> == __NOEDITSECTION__ <!-- Remove the section edit links --> ff27065f1e2d42507088ef553432bb109e57a09d 34 22 2022-09-05T20:50:53Z Metachirality 7 wikitext text/x-wiki Epsilon numbers are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). == <span id="zero">\(\varepsilon_{0}\)</span> == == <span id="one">\(\varepsilon_{1}\)</span> == == <span id="omega">\(\varepsilon_{\omega}\)</span> == == <span id="epsilonzero">\(\varepsilon_{\varepsilon_{0}}\)</span> == __NOEDITSECTION__ <!-- Remove the section edit links --> 13c3ab6af70849e4bd4289c510794a332f1bf0b3 35 34 2022-09-05T20:51:29Z Metachirality 7 wikitext text/x-wiki '''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). == <span id="zero">\(\varepsilon_{0}\)</span> == == <span id="one">\(\varepsilon_{1}\)</span> == == <span id="omega">\(\varepsilon_{\omega}\)</span> == == <span id="epsilonzero">\(\varepsilon_{\varepsilon_{0}}\)</span> == __NOEDITSECTION__ <!-- Remove the section edit links --> db02db4959b85843eed1c902d706d11029569015 36 35 2022-09-05T21:17:39Z Augigogigi 2 wikitext text/x-wiki '''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). __NOEDITSECTION__ <!-- Remove the section edit links --> 7991f5e9b0a5a1beea494a2b022617b34bf7aab6 2 12 23 2022-09-03T22:36:10Z Yto 4 Created page with "Hi, i've been an apeirologist for approximately 4 years. I thought about writing articles on this wiki, but realized that i'm bad at finding sources, so i'll just use this page for unsourced explanations and philosophy related to apeirology. Hopefully that's ok." wikitext text/x-wiki Hi, i've been an apeirologist for approximately 4 years. I thought about writing articles on this wiki, but realized that i'm bad at finding sources, so i'll just use this page for unsourced explanations and philosophy related to apeirology. Hopefully that's ok. b205eb9ca3280a5673a55afcad036391d42228f1 24 23 2022-09-03T23:40:21Z Yto 4 wikitext text/x-wiki Hi, i've been an apeirologist for approximately 4 years. I thought about writing articles on this wiki, but realized that i'm bad at finding sources, so i'll just use this page for unsourced explanations and philosophy related to apeirology. Hopefully that's ok. Social and formal proofs There are two kinds of proofs in math. Formal proofs start with specified axioms and use well-defined rules to make inferences. Meanwhile, social proofs don't necessarily start with any axioms, and inferences can be made intuitively. The goal of a social proof is to convince the reader that something is true, while the goal of a formal proof is to show that if the axioms are true, then the theorems are also true, therefore a social proof of the axioms would lead to a social proof of the theorems. I'll edit this to provide examples soon. Platonism Platonism is the philosophy of Plato. It asserts that abstract objects such as beauty and injustice exist. In the context of mathematics (mainly set theory), i think "platonism" commonly refers to a view asserting that something has a value independent of our thoughts. For example, a platonist may believe in an independent universe of sets, or only in an independent set of natural numbers. Due to this variety, platonism is not a single belief, but a way to compare beliefs - one belief is "more platonic" than another if it asserts that more things are independent of thoughts. A view with absolutely no platonism is pointless, because that would mean nothing can be certain, not even statements such as 1+1=2, or the existence of anything at all. For example, if we abandon the idea that there is no integer between 0 and 1, despite this being true in our physical world, there are suddenly many things we cannot be certain about. What if 7 is the largest prime number, and all the larger numbers that we call "primes" are actually divisible by this hidden integer n? How would a function with n inputs work? Is 2+2 still 4, or could it now be 3? Maybe 2+2 is an integer between 3 and 4. Being too unplatonic leads to confusing (but entertaining) questions like this, and if we want to get any advanced results, it's more useful to simply assume that some things are the way we expect them to be. This is closely related to social proofs. A more platonic view allows more social proofs. For example, the belief that the axioms of Peano arithmetic are true independently of our thoughts basically directly assigns social proofs to all formal proofs in Peano arithmetic. However, a belief in an independent set of natural numbers can lead to social proofs of much more, probably including things like the consistency of Peano arithmetic. It also implies that there are independent truth values of the twin prime conjecture and the goldbach conjecture, but we don't have any social proofs of those truth values yet, and i don't know whether it's possible that social proofs of these would require a more platonic view, or even not exist at all. I'll add more to this eventually. d08db310e5191e12a3fa8b32bb77057bee6c7f0a 25 24 2022-09-04T13:54:52Z Yto 4 wikitext text/x-wiki Hi, i've been an apeirologist for approximately 4 years. I thought about writing articles on this wiki, but realized that i'm bad at finding sources, so i'll just use this page for unsourced explanations and philosophy related to apeirology. Hopefully that's ok. Platonism 1. Introduction Platonism is the philosophy of Plato. It asserts that abstract objects such as beauty and injustice exist. In the context of mathematics (mainly set theory), i think "platonism" commonly refers to a view asserting that something has a value independent of our thoughts. For example, a platonist may believe in an independent universe of sets, or only in an independent set of natural numbers. Due to this variety, platonism is not a single belief, but a way to compare beliefs - one belief is "more platonic" than another if it asserts that more things are independent of thoughts. A view with absolutely no platonism is pointless, because that would mean nothing can be certain, not even statements such as 1+1=2, or the existence of anything at all. For example, if we abandon the idea that there is no integer between 0 and 1, despite this being true in our physical world, there are suddenly many things we cannot be certain about. What if 7 is the largest prime number, and all the larger numbers that we call "primes" are actually divisible by this hidden integer n? How would a function with n inputs work? Is 2+2 still 4, or could it now be 3? Maybe 2+2 is an integer between 3 and 4. Being too unplatonic leads to confusing (but entertaining) questions like this, and if we want to get any advanced results, it's more useful to simply assume that some things are the way we expect them to be. 2. Social and formal proofs There are two kinds of proofs in math. Formal proofs start with specified axioms and use well-defined rules to make inferences. Meanwhile, social proofs don't necessarily start with any axioms, and inferences can be made intuitively. The goal of a social proof is to convince the reader that something is true, while the goal of a formal proof is to show that if the axioms are true, then the theorems are also true, therefore a social proof of the axioms would lead to a social proof of the theorems. A perfect example of social proofs are visual proofs. This is closely related to platonism. A more platonic view allows more social proofs. For example, the belief that the axioms of Peano arithmetic are true independently of our thoughts basically directly assigns social proofs to all formal proofs in Peano arithmetic. However, a belief in an independent set of natural numbers can lead to social proofs of much more, probably including things like the consistency of Peano arithmetic. It also implies that there are independent truth values of the twin prime conjecture and the goldbach conjecture, but we don't have any social proofs of those truth values yet, and i don't know whether it's possible that social proofs of these would require a more platonic view, or even not exist at all. Of course, as i mentioned in the introduction, there are views so unplatonic that they allow the existence of integers between our usual integers, or perhaps even the non-existence of our usual integers. With these views, even formal proofs are in danger, as for example certain things that look like formal proofs to us may actually just not exist at all, or may contain inferences that look like they're allowed but aren't. This makes formal proofs also a little social, so the main distinction between social and formal proofs is that formal proofs can be verified by a computer. But again, while it can be entertaining to think this way, it's unlikely to be productive. 3. Different kinds of well-definedness For apeirology, platonism is relevant because the well-definedness of certain things may depend on what kinds of proofs are allowed, and this depends on the view. A view that isn't platonic enough can put an upper bound on the size of well-defined ordinals, or on the growth rate of well-defined functions for finite numbers. Weirdly enough, it also depends on what kind of well-definedness we're interested in. Well-definedness seems like a straightforward idea - something is well-defined if it has a unique value. We can, for example, easily prove the well-definedness of the busy beaver function - it's clear that for every n, there are finitely many halting n-state turing machines, so there's a finite set of numbers of ones left after halting, and we can take the maximum of that set. This can be proven formally. However, no matter what consistent recursive theory you work in, you can't prove each value of the busy beaver function constructively. Assuming ZFC is consistent, even if you knew the specific value of BB(800), you can't prove in ZFC that BB(800) is equal to that value, because there's a turing machine with 748 states that halts iff ZFC is inconsistent, and ZFC can't disprove that. Even if ZFC could prove that the machine can't halt with more than the claimed value of BB(800) ones on the tape, there probably is a simple way to prevent that with the 52 extra states. For stronger theories, we can just increase the input and the same thing will apply. So is the busy beaver function really well-defined? This is where the different kinds of well-definedness come in. There are at least 4 important ones when it comes to functions (i haven't yet thought about this for anything else, sorry), which may have actual names already, but i would have to look for sources so i just gave them some placeholder names for now: Nonconstructive individual (NI) well-definedness, Nonconstructive global (NG) well-definedness, Constructive individual (CI) well-definedness, and Constructive global (CG) well-definedness. A function f is NI well-defined in T iff for every x in f's domain, T can prove there's a unique y such that f(x)=y. A function f is NG well-defined in T iff T can prove that for every x in f's domain, there's a unique y such that f(x)=y. A function f is CI well-defined in T iff for every x in f's domain, there's a unique y such that T can prove f(x)=y. The only difference between these three is the position of "T can prove" in the definition, and it's an important difference. Anyway, onto the fourth kind of well-definedness: A function f is CG well-defined in T iff it's both NG and CI well-defined in T. It's worth noting that when T is a relatively nice theory, NG well-definedness in T implies NI well-definedness in T, and so does CI well-definedness in T, so these 4 kinds of well-definedness form a sort of square of implications. So if we allow only formal proofs, then all CI well-defined functions are computable. That excludes BB, which i've shown above, but also Rayo's function and every Kleene's O. There are even problems with NG well-definedness of Rayo's function, although those can be fixed by using a second-order theory (which still doesn't fix CI well-definedness). Now, this is just my opinion, but i think CG well-definedness is the main kind of well-definedness we should be interested in, and that is one of the biggest reasons why my view is very platonic - allowing social proofs makes many NG well-defined things also CG well-defined. I'll try to add more to this eventually. cb34ade2c1a8459604f32d7779f771edb9c4f3fc 2 13 27 2022-09-05T20:18:14Z Augigogigi 2 page created wikitext text/x-wiki sbocf is bla bla bla limit is bla bla bla == Definition == * \( T_{a}(b) = \) * \( \tau(a) = \) * \( \kappa(0) = \) == Fundamental Sequences == bla bla bla function: <div style="width:40%;"> \begin{align} [] : \text{S} \times \mathbb{N} \rightarrow& \text{S} \\ (\text{S},n) \mapsto& \text{S}[n] \end{align} </div> bla bla bla bla bla == Analysis == {| class="wikitable" |- ! Expression !! Shorthand |- | \( \tau(a) \) || \( \Omega_{a} \) |- | \( \tau(B+a) \) || \( M_{a} \) |- | \( \tau(B\cdot2+a) \) || \( N_{a} \) |- | \( \tau(B^{2}+a) \) || \( G_{a} \) |} {| class="wikitable" ! SbOCF !! Ordinal !! Xi !! BMS |- | \( T_{\tau(0)}(0) \) || \( 1 \) || - || \( (0) \) |- | \( T_{\tau(0)}(1) \) || \( \omega \) || - || \( (0)(1) \) |- | \( T_{\tau(0)}(2) \) || \( \omega^{2} \) || - || \( (0)(1)(1) \) |- | \( T_{\tau(0)}(T_{\tau(0)}(0)) \) || \( \omega^{\omega} \) || - || \( (0)(1)(2) \) |- | \( T_{\tau(0)}(T_{\tau(0)}(T_{\tau(0)}(0))) \) || \( \omega^{\omega^{\omega}} \) || - || \( (0)(1)(2)(3) \) |- | \( T_{\tau(0)}(\tau(0)) \) || \( \varepsilon_{0} \) || - || \( (0,0)(1,1) \) |- | \( T_{\tau(0)}(\tau(0)+1) \) || \( \varepsilon_{0}\cdot\omega \) || - || \( (0,0)(1,1)(1,0) \) |- | \( T_{\tau(0)}(\tau(0)+T_{\tau(0)}(0)) \) || \( \varepsilon_{0}\cdot\omega^{\omega} \) || - || \( (0,0)(1,1)(1,0)(2,0) \) |- | \( T_{\tau(0)}(\tau(0)+\tau(0)) \) || \( \varepsilon_{0}^{2} \) || - || \( (0,0)(1,1)(1,0)(2,1) \) |- | \( T_{\tau(0)}(\tau(0)\cdot\omega) \) || \( \varepsilon_{0}^{\omega} \) || - || \( (0,0)(1,1)(1,1) \) |} f4becd12b613e4231bbbf25d6419a7d6cb34cc3e 2 14 29 2022-09-05T20:31:14Z Augigogigi 2 Created page with "*\( \alpha \vartriangleleft_{0} A \) is true iff for all functions \( f : \alpha \rightarrow \alpha \), there exists \( \kappa \in A \) that is closed under \( f \) *\( [0]M = \{ \alpha : \alpha \vartriangleleft Ord \} \) *\( [S,\alpha + 1]\Xi_{0} = \{ \beta : \beta \vartriangleleft [\alpha]\Xi_{0} \} \) *\( [S,\alpha]\Xi_{0} = \bigcap_{\beta<\alpha} [\beta]\Xi_{0} \) iff \( \alpha \in Lim \) *\( [1,0,S]\Xi_{0} = \{ \alpha : \alpha = \bigcap_{\beta<\alpha} [\beta,S]\Xi_{..." wikitext text/x-wiki *\( \alpha \vartriangleleft_{0} A \) is true iff for all functions \( f : \alpha \rightarrow \alpha \), there exists \( \kappa \in A \) that is closed under \( f \) *\( [0]M = \{ \alpha : \alpha \vartriangleleft Ord \} \) *\( [S,\alpha + 1]\Xi_{0} = \{ \beta : \beta \vartriangleleft [\alpha]\Xi_{0} \} \) *\( [S,\alpha]\Xi_{0} = \bigcap_{\beta<\alpha} [\beta]\Xi_{0} \) iff \( \alpha \in Lim \) *\( [1,0,S]\Xi_{0} = \{ \alpha : \alpha = \bigcap_{\beta<\alpha} [\beta,S]\Xi_{0} \} \) *\( A(\alpha) = \text{enum}(A) \) *\( r(A,[\alpha]\Xi_{0}(S,\beta + 1,0)) = \{ \gamma : \gamma = [\alpha]\Xi_{0}(S,\beta,\gamma) \land \gamma \in A \} \) 703399ee43ade00b440ce8066dbe1015326df574 0 15 31 2022-09-05T20:37:16Z Metachirality 7 Created page with "'''Bashicu matrix system''' is an [[ordinal notation]] and term system invented by [[Bashicu]]." wikitext text/x-wiki '''Bashicu matrix system''' is an [[ordinal notation]] and term system invented by [[Bashicu]]. fc41b3775642583edca1ba8969e5eca2e3866f38 0 16 33 2022-09-05T20:48:13Z Metachirality 7 Created page with "A normal function is a function on [[ordinal|ordinals]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]." wikitext text/x-wiki A normal function is a function on [[ordinal|ordinals]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]. c1bf0375488188d43cbd0eb9d2d0543513b470c1 0 17 39 2022-09-05T21:26:33Z C7X 9 C7X moved page [[List of ordinals]] to [[List of countable ordinals]] wikitext text/x-wiki #REDIRECT [[List of countable ordinals]] c67ac5e710679ab2cb2d6fe0027dd882bd3bf1eb 40 39 2022-09-05T21:48:26Z Augigogigi 2 extend and clean up the list wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega*2|\( \omega\cdot2 \)]] * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] * [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] * [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \) * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \text{KPi} \) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \) * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE * STABILITY STUFF GOES HERE * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] ==== Uncountables ==== * [[omega_1|\( \Omega_{1} \), the smallest [[uncountable ordinal]] d0aac0bcba7ae33ee5407a2b60dd2b679686c41e 41 40 2022-09-05T21:49:17Z Augigogigi 2 wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega*2|\( \omega\cdot2 \)]] * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] * [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] * [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \) * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \text{KPi} \) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \) * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE * STABILITY STUFF GOES HERE * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] ==== Uncountables ==== * [[omega_1|\( \Omega_{1} \)]], the smallest [[uncountable ordinal]] 6f18360da4b474cd9b9211cf14adcdbfeee482e3 42 41 2022-09-05T21:56:56Z Augigogigi 2 wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega*2|\( \omega\cdot2 \)]] * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] * [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] * [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \) * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \text{KPi} \) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \) * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE * STABILITY STUFF GOES HERE * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] ==== Uncountables ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] edd1ffabb555bb072acd41303ae57a1b5e278864 43 42 2022-09-05T21:59:44Z Augigogigi 2 wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega*2|\( \omega\cdot2 \)]] * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] * [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] * [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of \( \textsf{ATR}_{0} \) * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \textsf{KPi} \) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \textsf{KPM} \) * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \textsf{KP} + \Pi_{3}\textsf{-refl.} \) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \textsf{KP} \) with a \( \Pi_{\mathbb{N}}\textsf{-refl.} \) universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE * STABILITY STUFF GOES HERE * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] ==== Uncountables ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] 2c47aa487ec866fd450edf10f3cd851216a9754f 44 43 2022-09-05T22:01:40Z Augigogigi 2 wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega*2|\( \omega\cdot2 \)]] * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] * [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] * [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \) * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \text{KPi} \) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \) * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE * STABILITY STUFF GOES HERE * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] ==== Uncountables ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] edd1ffabb555bb072acd41303ae57a1b5e278864 45 44 2022-09-05T22:09:28Z Augigogigi 2 wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * STABILITY STUFF GOES HERE^{(sort out)} * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] * MORE STUFF GOES HERE^{(sort out)} ==== Uncountables ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] 03b4691982d5da9318269ab14892cb022c00aa0b 46 45 2022-09-05T22:18:04Z Augigogigi 2 wikitext text/x-wiki ==== Countables ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * STABILITY STUFF GOES HERE^{(sort out)} * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] * MORE STUFF GOES HERE^{(sort out)} ==== Uncountables ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] b7f131409d9603ab41c1ed1c2dd702bd0d252039 0 18 47 2022-09-05T22:20:28Z Augigogigi 2 page created wikitext text/x-wiki * [[cantor_normal_form|Cantor normal form]], or CNF * The [[veblen_hierarchy|Veblen hierarchy]] * [[buchholz_psi|Buchholz's psi] * [[extended_buchholz_psi|Extended Buchholz's psi] * rathjen's ocfs^{(sort out)} * arai's ocfs^{(sort out)} * stegert's ocfs^{(sort out)} 5a415c5ac28d908c5a1144d61554148cc5dabb14 48 47 2022-09-05T22:21:25Z Augigogigi 2 wikitext text/x-wiki * [[cantor_normal_form|Cantor normal form]], or CNF * The [[veblen_hierarchy|Veblen hierarchy]] * [[buchholz_psi|Buchholz's psi]] * [[extended_buchholz_psi|Extended Buchholz's psi]] * rathjen's ocfs^{(sort out)} * arai's ocfs^{(sort out)} * stegert's ocfs^{(sort out)} e73482595a03b61dfacb8cf53ed52870acfcf1a2 1 19 49 2022-09-06T00:08:32Z Augigogigi 2 page created wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 50 49 2022-09-06T00:14:13Z Augigogigi 2 /* What should the page be */ new section wikitext text/x-wiki == What should the page be == I opened a vote in the discord as to what the page should contain, reposting here for documentation. the options: 🇦 a list of every ordinal with a page 🇧 a list of every ordinal deemed 'significant' under some definition of 'significant', with some of them not having pages 🇨 a 1000-steps-esque page with a LOT of ordinals, and bookmarks to the significant ones 🇩 something else (pls elaborate) 🇪 do C [List of Ordinals] and B [List of Significant Ordinals] awaiting results --[[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 00:14, 6 September 2022 (UTC) a16c03401a174947ca4c84ecba32f2c359dd911c 51 50 2022-09-06T00:14:53Z Augigogigi 2 wikitext text/x-wiki == What should the page be == I opened a vote in the discord as to what the page should contain, reposting here for documentation. the options: 🇦 a list of every ordinal with a page 🇧 a list of every ordinal deemed 'significant' under some definition of 'significant', with some of them not having pages 🇨 a 1000-steps-esque page with a LOT of ordinals, and bookmarks to the significant ones 🇩 something else (pls elaborate) 🇪 do C [List of Ordinals] and B [List of Significant Ordinals] awaiting results -- [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 00:14, 6 September 2022 (UTC) d1e9a8c67e59a9d79f167724a4e837a031a074fd 52 51 2022-09-06T00:17:05Z Augigogigi 2 wikitext text/x-wiki == What should the page be == I opened a vote in the discord as to what the page should contain, reposting here for documentation. the options: # a list of every ordinal with a page # a list of every ordinal deemed 'significant' under some definition of 'significant', with some of them not having pages # a 1000-steps-esque page with a LOT of ordinals, and bookmarks to the significant ones # something else (pls elaborate) # do C [List of Ordinals] and B [List of Significant Ordinals] <br> awaiting results <br> --→ [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 00:14, 6 September 2022 (UTC) 93f9a98fc332cd2626f68b661f7e4f4ab2125707 0 20 57 2022-09-06T02:13:56Z OfficialURL 10 create stub wikitext text/x-wiki In general mathematics, a '''fixed point''' of a function \(f:X\to X\) is any \(x\in X\) such that \(f(x)=x\). 2dae7ae33a800396baeb7103bfb2f62c9a99772f 10 21 58 2022-09-06T02:19:36Z OfficialURL 10 Copied from https://polytope.miraheze.org/w/index.php?title=Template:Wikipedia wikitext text/x-wiki <includeonly>Wikipedia Contributors. [https://en.wikipedia.org/wiki/{{ucfirst:{{replace|{{{1|{{PAGENAME}}}}}| |_}}}} "{{{2|{{ucfirst:{{{1|{{PAGENAME}}}}}}}}}}"].</includeonly> <noinclude>{{documentation}}</noinclude> 50f5088a5014ee1ff64314faa003555cb8454cd0 10 22 59 2022-09-06T02:20:21Z OfficialURL 10 Copied from https://polytope.miraheze.org/w/index.php?title=Template:Wikipedia/doc wikitext text/x-wiki ==Usage== <code><nowiki>{{Wikipedia|url|name}}</nowiki></code>links to <nowiki>https://en.wikipedia.org/wiki/</nowiki>''url'' (with spaces on the URL replaced by hyphens), and references the page name as ''name''. The second argument can be excluded when the URL and the implied page name coincide (most of the time). ==Examples== *<code><nowiki>{{Wikipedia|Tetrahedron}}</nowiki></code>→ {{Wikipedia|Tetrahedron}} *<code><nowiki>{{Wikipedia|Grand antiprism}}</nowiki></code>→ {{Wikipedia|Grand antiprism}} *<code><nowiki>{{Wikipedia|Truncated 5-cell#Bitruncated 5-cell|Bitruncated 5-cell}}</nowiki></code>→ {{Wikipedia|Truncated 5-cell#Bitruncated 5-cell|Bitruncated 5-cell}} [[Category:External resource templates]] b459a6ed8d80deac46ac4980a8acb0da7ce53904 10 23 60 2022-09-06T02:21:03Z OfficialURL 10 Created page with "{{#invoke:documentation|main|_content={{ {{#invoke:documentation|contentTitle}}}}}}<noinclude> <!-- Categories go on the /doc subpage, and interwikis go on Wikidata. --> </noinclude>" wikitext text/x-wiki {{#invoke:documentation|main|_content={{ {{#invoke:documentation|contentTitle}}}}}}<noinclude> <!-- Categories go on the /doc subpage, and interwikis go on Wikidata. --> </noinclude> ce7fd93f18c46b4fa871bf679afd05cbda72d8c4 828 24 61 2022-09-06T02:21:37Z OfficialURL 10 Created page with "-- This module implements {{documentation}}. -- Get required modules. local getArgs = require('Module:Arguments').getArgs local messageBox = require('Module:Message box') -- Get the config table. local cfg = mw.loadData('Module:Documentation/config') local p = {} -- Often-used functions. local ugsub = mw.ustring.gsub ---------------------------------------------------------------------------- -- Helper functions -- -- These are defined as local functions, but are ma..." Scribunto text/plain -- This module implements {{documentation}}. -- Get required modules. local getArgs = require('Module:Arguments').getArgs local messageBox = require('Module:Message box') -- Get the config table. local cfg = mw.loadData('Module:Documentation/config') local p = {} -- Often-used functions. local ugsub = mw.ustring.gsub ---------------------------------------------------------------------------- -- Helper functions -- -- These are defined as local functions, but are made available in the p -- table for testing purposes. ---------------------------------------------------------------------------- local function message(cfgKey, valArray, expectType) --[[ -- Gets a message from the cfg table and formats it if appropriate. -- The function raises an error if the value from the cfg table is not -- of the type expectType. The default type for expectType is 'string'. -- If the table valArray is present, strings such as $1, $2 etc. in the -- message are substituted with values from the table keys [1], [2] etc. -- For example, if the message "foo-message" had the value 'Foo $2 bar $1.', -- message('foo-message', {'baz', 'qux'}) would return "Foo qux bar baz." --]] local msg = cfg[cfgKey] expectType = expectType or 'string' if type(msg) ~= expectType then error('message: type error in message cfg.' .. cfgKey .. ' (' .. expectType .. ' expected, got ' .. type(msg) .. ')', 2) end if not valArray then return msg end local function getMessageVal(match) match = tonumber(match) return valArray[match] or error('message: no value found for key $' .. match .. ' in message cfg.' .. cfgKey, 4) end local ret = ugsub(msg, '$([1-9][0-9]*)', getMessageVal) return ret end p.message = message local function makeWikilink(page, display) if display then return mw.ustring.format('[[%s|%s]]', page, display) else return mw.ustring.format('[[%s]]', page) end end p.makeWikilink = makeWikilink local function makeCategoryLink(cat, sort) local catns = mw.site.namespaces[14].name return makeWikilink(catns .. ':' .. cat, sort) end p.makeCategoryLink = makeCategoryLink local function makeUrlLink(url, display) return mw.ustring.format('[%s %s]', url, display) end p.makeUrlLink = makeUrlLink local function makeToolbar(...) local ret = {} local lim = select('#', ...) if lim < 1 then return nil end for i = 1, lim do ret[#ret + 1] = select(i, ...) end return '<small style="font-style: normal;">(' .. table.concat(ret, ' &#124; ') .. ')</small>' end p.makeToolbar = makeToolbar ---------------------------------------------------------------------------- -- Argument processing ---------------------------------------------------------------------------- local function makeInvokeFunc(funcName) return function (frame) local args = getArgs(frame, { valueFunc = function (key, value) if type(value) == 'string' then value = value:match('^%s*(.-)%s*$') -- Remove whitespace. if key == 'heading' or value ~= '' then return value else return nil end else return value end end }) return p[funcName](args) end end ---------------------------------------------------------------------------- -- Main function ---------------------------------------------------------------------------- p.main = makeInvokeFunc('_main') function p._main(args) --[[ -- This function defines logic flow for the module. -- @args - table of arguments passed by the user -- -- Messages: -- 'main-div-id' --> 'template-documentation' -- 'main-div-classes' --> 'template-documentation iezoomfix' --]] local env = p.getEnvironment(args) local root = mw.html.create() root :wikitext(p.protectionTemplate(env)) :wikitext(p.sandboxNotice(args, env)) -- This div tag is from {{documentation/start box}}, but moving it here -- so that we don't have to worry about unclosed tags. :tag('div') :attr('id', message('main-div-id')) :addClass(message('main-div-classes')) :newline() :wikitext(p._startBox(args, env)) :wikitext(p._content(args, env)) :tag('div') :css('clear', 'both') -- So right or left floating items don't stick out of the doc box. :newline() :done() :done() :wikitext(p._endBox(args, env)) :wikitext(p.addTrackingCategories(env)) return tostring(root) end ---------------------------------------------------------------------------- -- Environment settings ---------------------------------------------------------------------------- function p.getEnvironment(args) --[[ -- Returns a table with information about the environment, including title objects and other namespace- or -- path-related data. -- @args - table of arguments passed by the user -- -- Title objects include: -- env.title - the page we are making documentation for (usually the current title) -- env.templateTitle - the template (or module, file, etc.) -- env.docTitle - the /doc subpage. -- env.sandboxTitle - the /sandbox subpage. -- env.testcasesTitle - the /testcases subpage. -- env.printTitle - the print version of the template, located at the /Print subpage. -- -- Data includes: -- env.protectionLevels - the protection levels table of the title object. -- env.subjectSpace - the number of the title's subject namespace. -- env.docSpace - the number of the namespace the title puts its documentation in. -- env.docpageBase - the text of the base page of the /doc, /sandbox and /testcases pages, with namespace. -- env.compareUrl - URL of the Special:ComparePages page comparing the sandbox with the template. -- -- All table lookups are passed through pcall so that errors are caught. If an error occurs, the value -- returned will be nil. --]] local env, envFuncs = {}, {} -- Set up the metatable. If triggered we call the corresponding function in the envFuncs table. The value -- returned by that function is memoized in the env table so that we don't call any of the functions -- more than once. (Nils won't be memoized.) setmetatable(env, { __index = function (t, key) local envFunc = envFuncs[key] if envFunc then local success, val = pcall(envFunc) if success then env[key] = val -- Memoise the value. return val end end return nil end }) function envFuncs.title() -- The title object for the current page, or a test page passed with args.page. local title local titleArg = args.page if titleArg then title = mw.title.new(titleArg) else title = mw.title.getCurrentTitle() end return title end function envFuncs.templateTitle() --[[ -- The template (or module, etc.) title object. -- Messages: -- 'sandbox-subpage' --> 'sandbox' -- 'testcases-subpage' --> 'testcases' --]] local subjectSpace = env.subjectSpace local title = env.title local subpage = title.subpageText if subpage == message('sandbox-subpage') or subpage == message('testcases-subpage') then return mw.title.makeTitle(subjectSpace, title.baseText) else return mw.title.makeTitle(subjectSpace, title.text) end end function envFuncs.docTitle() --[[ -- Title object of the /doc subpage. -- Messages: -- 'doc-subpage' --> 'doc' --]] local title = env.title local docname = args[1] -- User-specified doc page. local docpage if docname then docpage = docname else docpage = env.docpageBase .. '/' .. message('doc-subpage') end return mw.title.new(docpage) end function envFuncs.sandboxTitle() --[[ -- Title object for the /sandbox subpage. -- Messages: -- 'sandbox-subpage' --> 'sandbox' --]] return mw.title.new(env.docpageBase .. '/' .. message('sandbox-subpage')) end function envFuncs.testcasesTitle() --[[ -- Title object for the /testcases subpage. -- Messages: -- 'testcases-subpage' --> 'testcases' --]] return mw.title.new(env.docpageBase .. '/' .. message('testcases-subpage')) end function envFuncs.printTitle() --[[ -- Title object for the /Print subpage. -- Messages: -- 'print-subpage' --> 'Print' --]] return env.templateTitle:subPageTitle(message('print-subpage')) end function envFuncs.protectionLevels() -- The protection levels table of the title object. return env.title.protectionLevels end function envFuncs.subjectSpace() -- The subject namespace number. return mw.site.namespaces[env.title.namespace].subject.id end function envFuncs.docSpace() -- The documentation namespace number. For most namespaces this is the same as the -- subject namespace. However, pages in the Article, File, MediaWiki or Category -- namespaces must have their /doc, /sandbox and /testcases pages in talk space. local subjectSpace = env.subjectSpace if subjectSpace == 0 or subjectSpace == 6 or subjectSpace == 8 or subjectSpace == 14 then return subjectSpace + 1 else return subjectSpace end end function envFuncs.docpageBase() -- The base page of the /doc, /sandbox, and /testcases subpages. -- For some namespaces this is the talk page, rather than the template page. local templateTitle = env.templateTitle local docSpace = env.docSpace local docSpaceText = mw.site.namespaces[docSpace].name -- Assemble the link. docSpace is never the main namespace, so we can hardcode the colon. return docSpaceText .. ':' .. templateTitle.text end function envFuncs.compareUrl() -- Diff link between the sandbox and the main template using [[Special:ComparePages]]. local templateTitle = env.templateTitle local sandboxTitle = env.sandboxTitle if templateTitle.exists and sandboxTitle.exists then local compareUrl = mw.uri.fullUrl( 'Special:ComparePages', {page1 = templateTitle.prefixedText, page2 = sandboxTitle.prefixedText} ) return tostring(compareUrl) else return nil end end return env end ---------------------------------------------------------------------------- -- Auxiliary templates ---------------------------------------------------------------------------- function p.sandboxNotice(args, env) --[=[ -- Generates a sandbox notice for display above sandbox pages. -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- Messages: -- 'sandbox-notice-image' --> '[[Image:Sandbox.svg|50px|alt=|link=]]' -- 'sandbox-notice-blurb' --> 'This is the $1 for $2.' -- 'sandbox-notice-diff-blurb' --> 'This is the $1 for $2 ($3).' -- 'sandbox-notice-pagetype-template' --> '[[Wikipedia:Template test cases|template sandbox]] page' -- 'sandbox-notice-pagetype-module' --> '[[Wikipedia:Template test cases|module sandbox]] page' -- 'sandbox-notice-pagetype-other' --> 'sandbox page' -- 'sandbox-notice-compare-link-display' --> 'diff' -- 'sandbox-notice-testcases-blurb' --> 'See also the companion subpage for $1.' -- 'sandbox-notice-testcases-link-display' --> 'test cases' -- 'sandbox-category' --> 'Template sandboxes' --]=] local title = env.title local sandboxTitle = env.sandboxTitle local templateTitle = env.templateTitle local subjectSpace = env.subjectSpace if not (subjectSpace and title and sandboxTitle and templateTitle and mw.title.equals(title, sandboxTitle)) then return nil end -- Build the table of arguments to pass to {{ombox}}. We need just two fields, "image" and "text". local omargs = {} omargs.image = message('sandbox-notice-image') -- Get the text. We start with the opening blurb, which is something like -- "This is the template sandbox for [[Template:Foo]] (diff)." local text = '' local pagetype if subjectSpace == 10 then pagetype = message('sandbox-notice-pagetype-template') elseif subjectSpace == 828 then pagetype = message('sandbox-notice-pagetype-module') else pagetype = message('sandbox-notice-pagetype-other') end local templateLink = makeWikilink(templateTitle.prefixedText) local compareUrl = env.compareUrl if compareUrl then local compareDisplay = message('sandbox-notice-compare-link-display') local compareLink = makeUrlLink(compareUrl, compareDisplay) text = text .. message('sandbox-notice-diff-blurb', {pagetype, templateLink, compareLink}) else text = text .. message('sandbox-notice-blurb', {pagetype, templateLink}) end -- Get the test cases page blurb if the page exists. This is something like -- "See also the companion subpage for [[Template:Foo/testcases|test cases]]." local testcasesTitle = env.testcasesTitle if testcasesTitle and testcasesTitle.exists then if testcasesTitle.contentModel == "Scribunto" then local testcasesLinkDisplay = message('sandbox-notice-testcases-link-display') local testcasesRunLinkDisplay = message('sandbox-notice-testcases-run-link-display') local testcasesLink = makeWikilink(testcasesTitle.prefixedText, testcasesLinkDisplay) local testcasesRunLink = makeWikilink(testcasesTitle.talkPageTitle.prefixedText, testcasesRunLinkDisplay) text = text .. '<br />' .. message('sandbox-notice-testcases-run-blurb', {testcasesLink, testcasesRunLink}) else local testcasesLinkDisplay = message('sandbox-notice-testcases-link-display') local testcasesLink = makeWikilink(testcasesTitle.prefixedText, testcasesLinkDisplay) text = text .. '<br />' .. message('sandbox-notice-testcases-blurb', {testcasesLink}) end end -- Add the sandbox to the sandbox category. text = text .. makeCategoryLink(message('sandbox-category')) omargs.text = text local ret = '<div style="clear: both;"></div>' ret = ret .. messageBox.main('ombox', omargs) return ret end function p.protectionTemplate(env) -- Generates the padlock icon in the top right. -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- Messages: -- 'protection-template' --> 'pp-template' -- 'protection-template-args' --> {docusage = 'yes'} local protectionLevels, mProtectionBanner local title = env.title protectionLevels = env.protectionLevels if not protectionLevels then return nil end local editProt = protectionLevels.edit and protectionLevels.edit[1] local moveProt = protectionLevels.move and protectionLevels.move[1] if editProt then -- The page is edit-protected. mProtectionBanner = require('Module:Protection banner') local reason = message('protection-reason-edit') return mProtectionBanner._main{reason, small = true} elseif moveProt and moveProt ~= 'autoconfirmed' then -- The page is move-protected but not edit-protected. Exclude move -- protection with the level "autoconfirmed", as this is equivalent to -- no move protection at all. mProtectionBanner = require('Module:Protection banner') return mProtectionBanner._main{action = 'move', small = true} else return nil end end ---------------------------------------------------------------------------- -- Start box ---------------------------------------------------------------------------- p.startBox = makeInvokeFunc('_startBox') function p._startBox(args, env) --[[ -- This function generates the start box. -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- The actual work is done by p.makeStartBoxLinksData and p.renderStartBoxLinks which make -- the [view] [edit] [history] [purge] links, and by p.makeStartBoxData and p.renderStartBox -- which generate the box HTML. --]] env = env or p.getEnvironment(args) local links local content = args.content if not content or args[1] then -- No need to include the links if the documentation is on the template page itself. local linksData = p.makeStartBoxLinksData(args, env) if linksData then links = p.renderStartBoxLinks(linksData) end end -- Generate the start box html. local data = p.makeStartBoxData(args, env, links) if data then return p.renderStartBox(data) else -- User specified no heading. return nil end end function p.makeStartBoxLinksData(args, env) --[[ -- Does initial processing of data to make the [view] [edit] [history] [purge] links. -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- Messages: -- 'view-link-display' --> 'view' -- 'edit-link-display' --> 'edit' -- 'history-link-display' --> 'history' -- 'purge-link-display' --> 'purge' -- 'file-docpage-preload' --> 'Template:Documentation/preload-filespace' -- 'module-preload' --> 'Template:Documentation/preload-module-doc' -- 'docpage-preload' --> 'Template:Documentation/preload' -- 'create-link-display' --> 'create' --]] local subjectSpace = env.subjectSpace local title = env.title local docTitle = env.docTitle if not title or not docTitle then return nil end if docTitle.isRedirect then docTitle = docTitle.redirectTarget end local data = {} data.title = title data.docTitle = docTitle -- View, display, edit, and purge links if /doc exists. data.viewLinkDisplay = message('view-link-display') data.editLinkDisplay = message('edit-link-display') data.historyLinkDisplay = message('history-link-display') data.purgeLinkDisplay = message('purge-link-display') -- Create link if /doc doesn't exist. local preload = args.preload if not preload then if subjectSpace == 6 then -- File namespace preload = message('file-docpage-preload') elseif subjectSpace == 828 then -- Module namespace preload = message('module-preload') else preload = message('docpage-preload') end end data.preload = preload data.createLinkDisplay = message('create-link-display') return data end function p.renderStartBoxLinks(data) --[[ -- Generates the [view][edit][history][purge] or [create] links from the data table. -- @data - a table of data generated by p.makeStartBoxLinksData --]] local function escapeBrackets(s) -- Escapes square brackets with HTML entities. s = s:gsub('%[', '&#91;') -- Replace square brackets with HTML entities. s = s:gsub('%]', '&#93;') return s end local ret local docTitle = data.docTitle local title = data.title if docTitle.exists then local viewLink = makeWikilink(docTitle.prefixedText, data.viewLinkDisplay) local editLink = makeUrlLink(docTitle:fullUrl{action = 'edit'}, data.editLinkDisplay) local historyLink = makeUrlLink(docTitle:fullUrl{action = 'history'}, data.historyLinkDisplay) local purgeLink = makeUrlLink(title:fullUrl{action = 'purge'}, data.purgeLinkDisplay) ret = '[%s] [%s] [%s] [%s]' ret = escapeBrackets(ret) ret = mw.ustring.format(ret, viewLink, editLink, historyLink, purgeLink) else local createLink = makeUrlLink(docTitle:fullUrl{action = 'edit', preload = data.preload}, data.createLinkDisplay) ret = '[%s]' ret = escapeBrackets(ret) ret = mw.ustring.format(ret, createLink) end return ret end function p.makeStartBoxData(args, env, links) --[=[ -- Does initial processing of data to pass to the start-box render function, p.renderStartBox. -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- @links - a string containing the [view][edit][history][purge] links - could be nil if there's an error. -- -- Messages: -- 'documentation-icon-wikitext' --> '[[File:Test Template Info-Icon - Version (2).svg|50px|link=|alt=]]' -- 'template-namespace-heading' --> 'Template documentation' -- 'module-namespace-heading' --> 'Module documentation' -- 'file-namespace-heading' --> 'Summary' -- 'other-namespaces-heading' --> 'Documentation' -- 'start-box-linkclasses' --> 'mw-editsection-like plainlinks' -- 'start-box-link-id' --> 'doc_editlinks' -- 'testcases-create-link-display' --> 'create' --]=] local subjectSpace = env.subjectSpace if not subjectSpace then -- Default to an "other namespaces" namespace, so that we get at least some output -- if an error occurs. subjectSpace = 2 end local data = {} -- Heading local heading = args.heading -- Blank values are not removed. if heading == '' then -- Don't display the start box if the heading arg is defined but blank. return nil end if heading then data.heading = heading elseif subjectSpace == 10 then -- Template namespace data.heading = message('documentation-icon-wikitext') .. ' ' .. message('template-namespace-heading') elseif subjectSpace == 828 then -- Module namespace data.heading = message('documentation-icon-wikitext') .. ' ' .. message('module-namespace-heading') elseif subjectSpace == 6 then -- File namespace data.heading = message('file-namespace-heading') else data.heading = message('other-namespaces-heading') end -- Heading CSS local headingStyle = args['heading-style'] if headingStyle then data.headingStyleText = headingStyle elseif subjectSpace == 10 then -- We are in the template or template talk namespaces. data.headingFontWeight = 'bold' data.headingFontSize = '125%' else data.headingFontSize = '150%' end -- Data for the [view][edit][history][purge] or [create] links. if links then data.linksClass = message('start-box-linkclasses') data.linksId = message('start-box-link-id') data.links = links end return data end function p.renderStartBox(data) -- Renders the start box html. -- @data - a table of data generated by p.makeStartBoxData. local sbox = mw.html.create('div') sbox :css('padding-bottom', '3px') :css('border-bottom', '1px solid #aaa') :css('margin-bottom', '1ex') :newline() :tag('span') :cssText(data.headingStyleText) :css('font-weight', data.headingFontWeight) :css('font-size', data.headingFontSize) :wikitext(data.heading) local links = data.links if links then sbox:tag('span') :addClass(data.linksClass) :attr('id', data.linksId) :wikitext(links) end return tostring(sbox) end ---------------------------------------------------------------------------- -- Documentation content ---------------------------------------------------------------------------- p.content = makeInvokeFunc('_content') function p._content(args, env) -- Displays the documentation contents -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment env = env or p.getEnvironment(args) local docTitle = env.docTitle local content = args.content if not content and docTitle and docTitle.exists then content = args._content or mw.getCurrentFrame():expandTemplate{title = docTitle.prefixedText} end -- The line breaks below are necessary so that "=== Headings ===" at the start and end -- of docs are interpreted correctly. return '\n' .. (content or '') .. '\n' end p.contentTitle = makeInvokeFunc('_contentTitle') function p._contentTitle(args, env) env = env or p.getEnvironment(args) local docTitle = env.docTitle if not args.content and docTitle and docTitle.exists then return docTitle.prefixedText else return '' end end ---------------------------------------------------------------------------- -- End box ---------------------------------------------------------------------------- p.endBox = makeInvokeFunc('_endBox') function p._endBox(args, env) --[=[ -- This function generates the end box (also known as the link box). -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- Messages: -- 'fmbox-id' --> 'documentation-meta-data' -- 'fmbox-style' --> 'background-color: #ecfcf4' -- 'fmbox-textstyle' --> 'font-style: italic' -- -- The HTML is generated by the {{fmbox}} template, courtesy of [[Module:Message box]]. --]=] -- Get environment data. env = env or p.getEnvironment(args) local subjectSpace = env.subjectSpace local docTitle = env.docTitle if not subjectSpace or not docTitle then return nil end -- Check whether we should output the end box at all. Add the end -- box by default if the documentation exists or if we are in the -- user, module or template namespaces. local linkBox = args['link box'] if linkBox == 'off' or not ( docTitle.exists or subjectSpace == 2 or subjectSpace == 828 or subjectSpace == 10 ) then return nil end -- Assemble the arguments for {{fmbox}}. local fmargs = {} fmargs.id = message('fmbox-id') -- Sets 'documentation-meta-data' fmargs.image = 'none' fmargs.style = message('fmbox-style') -- Sets 'background-color: #ecfcf4' fmargs.textstyle = message('fmbox-textstyle') -- 'font-style: italic;' -- Assemble the fmbox text field. local text = '' if linkBox then text = text .. linkBox else text = text .. (p.makeDocPageBlurb(args, env) or '') -- "This documentation is transcluded from [[Foo]]." if subjectSpace == 2 or subjectSpace == 10 or subjectSpace == 828 then -- We are in the user, template or module namespaces. -- Add sandbox and testcases links. -- "Editors can experiment in this template's sandbox and testcases pages." text = text .. (p.makeExperimentBlurb(args, env) or '') text = text .. '<br />' if not args.content and not args[1] then -- "Please add categories to the /doc subpage." -- Don't show this message with inline docs or with an explicitly specified doc page, -- as then it is unclear where to add the categories. text = text .. (p.makeCategoriesBlurb(args, env) or '') end text = text .. ' ' .. (p.makeSubpagesBlurb(args, env) or '') --"Subpages of this template" local printBlurb = p.makePrintBlurb(args, env) -- Two-line blurb about print versions of templates. if printBlurb then text = text .. '<br />' .. printBlurb end end end fmargs.text = text return messageBox.main('fmbox', fmargs) end function p.makeDocPageBlurb(args, env) --[=[ -- Makes the blurb "This documentation is transcluded from [[Template:Foo]] (edit, history)". -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- Messages: -- 'edit-link-display' --> 'edit' -- 'history-link-display' --> 'history' -- 'transcluded-from-blurb' --> -- 'The above [[Wikipedia:Template documentation|documentation]] -- is [[Wikipedia:Transclusion|transcluded]] from $1.' -- 'module-preload' --> 'Template:Documentation/preload-module-doc' -- 'create-link-display' --> 'create' -- 'create-module-doc-blurb' --> -- 'You might want to $1 a documentation page for this [[Wikipedia:Lua|Scribunto module]].' --]=] local docTitle = env.docTitle if not docTitle then return nil end local ret if docTitle.exists then -- /doc exists; link to it. local docLink = makeWikilink(docTitle.prefixedText) local editUrl = docTitle:fullUrl{action = 'edit'} local editDisplay = message('edit-link-display') local editLink = makeUrlLink(editUrl, editDisplay) local historyUrl = docTitle:fullUrl{action = 'history'} local historyDisplay = message('history-link-display') local historyLink = makeUrlLink(historyUrl, historyDisplay) ret = message('transcluded-from-blurb', {docLink}) .. ' ' .. makeToolbar(editLink, historyLink) .. '<br />' elseif env.subjectSpace == 828 then -- /doc does not exist; ask to create it. local createUrl = docTitle:fullUrl{action = 'edit', preload = message('module-preload')} local createDisplay = message('create-link-display') local createLink = makeUrlLink(createUrl, createDisplay) ret = message('create-module-doc-blurb', {createLink}) .. '<br />' end return ret end function p.makeExperimentBlurb(args, env) --[[ -- Renders the text "Editors can experiment in this template's sandbox (edit | diff) and testcases (edit) pages." -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- Messages: -- 'sandbox-link-display' --> 'sandbox' -- 'sandbox-edit-link-display' --> 'edit' -- 'compare-link-display' --> 'diff' -- 'module-sandbox-preload' --> 'Template:Documentation/preload-module-sandbox' -- 'template-sandbox-preload' --> 'Template:Documentation/preload-sandbox' -- 'sandbox-create-link-display' --> 'create' -- 'mirror-edit-summary' --> 'Create sandbox version of $1' -- 'mirror-link-display' --> 'mirror' -- 'mirror-link-preload' --> 'Template:Documentation/mirror' -- 'sandbox-link-display' --> 'sandbox' -- 'testcases-link-display' --> 'testcases' -- 'testcases-edit-link-display'--> 'edit' -- 'template-sandbox-preload' --> 'Template:Documentation/preload-sandbox' -- 'testcases-create-link-display' --> 'create' -- 'testcases-link-display' --> 'testcases' -- 'testcases-edit-link-display' --> 'edit' -- 'module-testcases-preload' --> 'Template:Documentation/preload-module-testcases' -- 'template-testcases-preload' --> 'Template:Documentation/preload-testcases' -- 'experiment-blurb-module' --> 'Editors can experiment in this module's $1 and $2 pages.' -- 'experiment-blurb-template' --> 'Editors can experiment in this template's $1 and $2 pages.' --]] local subjectSpace = env.subjectSpace local templateTitle = env.templateTitle local sandboxTitle = env.sandboxTitle local testcasesTitle = env.testcasesTitle local templatePage = templateTitle.prefixedText if not subjectSpace or not templateTitle or not sandboxTitle or not testcasesTitle then return nil end -- Make links. local sandboxLinks, testcasesLinks if sandboxTitle.exists then local sandboxPage = sandboxTitle.prefixedText local sandboxDisplay = message('sandbox-link-display') local sandboxLink = makeWikilink(sandboxPage, sandboxDisplay) local sandboxEditUrl = sandboxTitle:fullUrl{action = 'edit'} local sandboxEditDisplay = message('sandbox-edit-link-display') local sandboxEditLink = makeUrlLink(sandboxEditUrl, sandboxEditDisplay) local compareUrl = env.compareUrl local compareLink if compareUrl then local compareDisplay = message('compare-link-display') compareLink = makeUrlLink(compareUrl, compareDisplay) end sandboxLinks = sandboxLink .. ' ' .. makeToolbar(sandboxEditLink, compareLink) else local sandboxPreload if subjectSpace == 828 then sandboxPreload = message('module-sandbox-preload') else sandboxPreload = message('template-sandbox-preload') end local sandboxCreateUrl = sandboxTitle:fullUrl{action = 'edit', preload = sandboxPreload} local sandboxCreateDisplay = message('sandbox-create-link-display') local sandboxCreateLink = makeUrlLink(sandboxCreateUrl, sandboxCreateDisplay) local mirrorSummary = message('mirror-edit-summary', {makeWikilink(templatePage)}) local mirrorPreload = message('mirror-link-preload') local mirrorUrl = sandboxTitle:fullUrl{action = 'edit', preload = mirrorPreload, summary = mirrorSummary} if subjectSpace == 828 then mirrorUrl = sandboxTitle:fullUrl{action = 'edit', preload = templateTitle.prefixedText, summary = mirrorSummary} end local mirrorDisplay = message('mirror-link-display') local mirrorLink = makeUrlLink(mirrorUrl, mirrorDisplay) sandboxLinks = message('sandbox-link-display') .. ' ' .. makeToolbar(sandboxCreateLink, mirrorLink) end if testcasesTitle.exists then local testcasesPage = testcasesTitle.prefixedText local testcasesDisplay = message('testcases-link-display') local testcasesLink = makeWikilink(testcasesPage, testcasesDisplay) local testcasesEditUrl = testcasesTitle:fullUrl{action = 'edit'} local testcasesEditDisplay = message('testcases-edit-link-display') local testcasesEditLink = makeUrlLink(testcasesEditUrl, testcasesEditDisplay) -- for Modules, add testcases run link if exists if testcasesTitle.contentModel == "Scribunto" and testcasesTitle.talkPageTitle and testcasesTitle.talkPageTitle.exists then local testcasesRunLinkDisplay = message('testcases-run-link-display') local testcasesRunLink = makeWikilink(testcasesTitle.talkPageTitle.prefixedText, testcasesRunLinkDisplay) testcasesLinks = testcasesLink .. ' ' .. makeToolbar(testcasesEditLink, testcasesRunLink) else testcasesLinks = testcasesLink .. ' ' .. makeToolbar(testcasesEditLink) end else local testcasesPreload if subjectSpace == 828 then testcasesPreload = message('module-testcases-preload') else testcasesPreload = message('template-testcases-preload') end local testcasesCreateUrl = testcasesTitle:fullUrl{action = 'edit', preload = testcasesPreload} local testcasesCreateDisplay = message('testcases-create-link-display') local testcasesCreateLink = makeUrlLink(testcasesCreateUrl, testcasesCreateDisplay) testcasesLinks = message('testcases-link-display') .. ' ' .. makeToolbar(testcasesCreateLink) end local messageName if subjectSpace == 828 then messageName = 'experiment-blurb-module' else messageName = 'experiment-blurb-template' end return message(messageName, {sandboxLinks, testcasesLinks}) end function p.makeCategoriesBlurb(args, env) --[[ -- Generates the text "Please add categories to the /doc subpage." -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- Messages: -- 'doc-link-display' --> '/doc' -- 'add-categories-blurb' --> 'Please add categories to the $1 subpage.' --]] local docTitle = env.docTitle if not docTitle then return nil end local docPathLink = makeWikilink(docTitle.prefixedText, message('doc-link-display')) return message('add-categories-blurb', {docPathLink}) end function p.makeSubpagesBlurb(args, env) --[[ -- Generates the "Subpages of this template" link. -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- Messages: -- 'template-pagetype' --> 'template' -- 'module-pagetype' --> 'module' -- 'default-pagetype' --> 'page' -- 'subpages-link-display' --> 'Subpages of this $1' --]] local subjectSpace = env.subjectSpace local templateTitle = env.templateTitle if not subjectSpace or not templateTitle then return nil end local pagetype if subjectSpace == 10 then pagetype = message('template-pagetype') elseif subjectSpace == 828 then pagetype = message('module-pagetype') else pagetype = message('default-pagetype') end local subpagesLink = makeWikilink( 'Special:PrefixIndex/' .. templateTitle.prefixedText .. '/', message('subpages-link-display', {pagetype}) ) return message('subpages-blurb', {subpagesLink}) end function p.makePrintBlurb(args, env) --[=[ -- Generates the blurb displayed when there is a print version of the template available. -- @args - a table of arguments passed by the user -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- -- Messages: -- 'print-link-display' --> '/Print' -- 'print-blurb' --> 'A [[Help:Books/for experts#Improving the book layout|print version]]' -- .. ' of this template exists at $1.' -- .. ' If you make a change to this template, please update the print version as well.' -- 'display-print-category' --> true -- 'print-category' --> 'Templates with print versions' --]=] local printTitle = env.printTitle if not printTitle then return nil end local ret if printTitle.exists then local printLink = makeWikilink(printTitle.prefixedText, message('print-link-display')) ret = message('print-blurb', {printLink}) local displayPrintCategory = message('display-print-category', nil, 'boolean') if displayPrintCategory then ret = ret .. makeCategoryLink(message('print-category')) end end return ret end ---------------------------------------------------------------------------- -- Tracking categories ---------------------------------------------------------------------------- function p.addTrackingCategories(env) --[[ -- Check if {{documentation}} is transcluded on a /doc or /testcases page. -- @env - environment table containing title objects, etc., generated with p.getEnvironment -- Messages: -- 'display-strange-usage-category' --> true -- 'doc-subpage' --> 'doc' -- 'testcases-subpage' --> 'testcases' -- 'strange-usage-category' --> 'Wikipedia pages with strange ((documentation)) usage' -- -- /testcases pages in the module namespace are not categorised, as they may have -- {{documentation}} transcluded automatically. --]] local title = env.title local subjectSpace = env.subjectSpace if not title or not subjectSpace then return nil end local subpage = title.subpageText local ret = '' if message('display-strange-usage-category', nil, 'boolean') and ( subpage == message('doc-subpage') or subjectSpace ~= 828 and subpage == message('testcases-subpage') ) then ret = ret .. makeCategoryLink(message('strange-usage-category')) end return ret end return p 02dd358f7f100661fa71458d29fd0513baadcfcc 828 25 62 2022-09-06T02:22:13Z OfficialURL 10 Created page with "-- This module provides easy processing of arguments passed to Scribunto from -- #invoke. It is intended for use by other Lua modules, and should not be -- called from #invoke directly. local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local arguments = {} -- Generate four different tidyVal functions, so that we don't have to check the -- options every time we call it. local function tidyValDefault(key, val) if type(val) == 'string'..." Scribunto text/plain -- This module provides easy processing of arguments passed to Scribunto from -- #invoke. It is intended for use by other Lua modules, and should not be -- called from #invoke directly. local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local arguments = {} -- Generate four different tidyVal functions, so that we don't have to check the -- options every time we call it. local function tidyValDefault(key, val) if type(val) == 'string' then val = val:match('^%s*(.-)%s*$') if val == '' then return nil else return val end else return val end end local function tidyValTrimOnly(key, val) if type(val) == 'string' then return val:match('^%s*(.-)%s*$') else return val end end local function tidyValRemoveBlanksOnly(key, val) if type(val) == 'string' then if val:find('%S') then return val else return nil end else return val end end local function tidyValNoChange(key, val) return val end local function matchesTitle(given, title) local tp = type( given ) return (tp == 'string' or tp == 'number') and mw.title.new( given ).prefixedText == title end local translate_mt = { __index = function(t, k) return k end } function arguments.getArgs(frame, options) checkType('getArgs', 1, frame, 'table', true) checkType('getArgs', 2, options, 'table', true) frame = frame or {} options = options or {} --[[ -- Set up argument translation. --]] options.translate = options.translate or {} if getmetatable(options.translate) == nil then setmetatable(options.translate, translate_mt) end if options.backtranslate == nil then options.backtranslate = {} for k,v in pairs(options.translate) do options.backtranslate[v] = k end end if options.backtranslate and getmetatable(options.backtranslate) == nil then setmetatable(options.backtranslate, { __index = function(t, k) if options.translate[k] ~= k then return nil else return k end end }) end --[[ -- Get the argument tables. If we were passed a valid frame object, get the -- frame arguments (fargs) and the parent frame arguments (pargs), depending -- on the options set and on the parent frame's availability. If we weren't -- passed a valid frame object, we are being called from another Lua module -- or from the debug console, so assume that we were passed a table of args -- directly, and assign it to a new variable (luaArgs). --]] local fargs, pargs, luaArgs if type(frame.args) == 'table' and type(frame.getParent) == 'function' then if options.wrappers then --[[ -- The wrappers option makes Module:Arguments look up arguments in -- either the frame argument table or the parent argument table, but -- not both. This means that users can use either the #invoke syntax -- or a wrapper template without the loss of performance associated -- with looking arguments up in both the frame and the parent frame. -- Module:Arguments will look up arguments in the parent frame -- if it finds the parent frame's title in options.wrapper; -- otherwise it will look up arguments in the frame object passed -- to getArgs. --]] local parent = frame:getParent() if not parent then fargs = frame.args else local title = parent:getTitle():gsub('/sandbox$', '') local found = false if matchesTitle(options.wrappers, title) then found = true elseif type(options.wrappers) == 'table' then for _,v in pairs(options.wrappers) do if matchesTitle(v, title) then found = true break end end end -- We test for false specifically here so that nil (the default) acts like true. if found or options.frameOnly == false then pargs = parent.args end if not found or options.parentOnly == false then fargs = frame.args end end else -- options.wrapper isn't set, so check the other options. if not options.parentOnly then fargs = frame.args end if not options.frameOnly then local parent = frame:getParent() pargs = parent and parent.args or nil end end if options.parentFirst then fargs, pargs = pargs, fargs end else luaArgs = frame end -- Set the order of precedence of the argument tables. If the variables are -- nil, nothing will be added to the table, which is how we avoid clashes -- between the frame/parent args and the Lua args. local argTables = {fargs} argTables[#argTables + 1] = pargs argTables[#argTables + 1] = luaArgs --[[ -- Generate the tidyVal function. If it has been specified by the user, we -- use that; if not, we choose one of four functions depending on the -- options chosen. This is so that we don't have to call the options table -- every time the function is called. --]] local tidyVal = options.valueFunc if tidyVal then if type(tidyVal) ~= 'function' then error( "bad value assigned to option 'valueFunc'" .. '(function expected, got ' .. type(tidyVal) .. ')', 2 ) end elseif options.trim ~= false then if options.removeBlanks ~= false then tidyVal = tidyValDefault else tidyVal = tidyValTrimOnly end else if options.removeBlanks ~= false then tidyVal = tidyValRemoveBlanksOnly else tidyVal = tidyValNoChange end end --[[ -- Set up the args, metaArgs and nilArgs tables. args will be the one -- accessed from functions, and metaArgs will hold the actual arguments. Nil -- arguments are memoized in nilArgs, and the metatable connects all of them -- together. --]] local args, metaArgs, nilArgs, metatable = {}, {}, {}, {} setmetatable(args, metatable) local function mergeArgs(tables) --[[ -- Accepts multiple tables as input and merges their keys and values -- into one table. If a value is already present it is not overwritten; -- tables listed earlier have precedence. We are also memoizing nil -- values, which can be overwritten if they are 's' (soft). --]] for _, t in ipairs(tables) do for key, val in pairs(t) do if metaArgs[key] == nil and nilArgs[key] ~= 'h' then local tidiedVal = tidyVal(key, val) if tidiedVal == nil then nilArgs[key] = 's' else metaArgs[key] = tidiedVal end end end end end --[[ -- Define metatable behaviour. Arguments are memoized in the metaArgs table, -- and are only fetched from the argument tables once. Fetching arguments -- from the argument tables is the most resource-intensive step in this -- module, so we try and avoid it where possible. For this reason, nil -- arguments are also memoized, in the nilArgs table. Also, we keep a record -- in the metatable of when pairs and ipairs have been called, so we do not -- run pairs and ipairs on the argument tables more than once. We also do -- not run ipairs on fargs and pargs if pairs has already been run, as all -- the arguments will already have been copied over. --]] metatable.__index = function (t, key) --[[ -- Fetches an argument when the args table is indexed. First we check -- to see if the value is memoized, and if not we try and fetch it from -- the argument tables. When we check memoization, we need to check -- metaArgs before nilArgs, as both can be non-nil at the same time. -- If the argument is not present in metaArgs, we also check whether -- pairs has been run yet. If pairs has already been run, we return nil. -- This is because all the arguments will have already been copied into -- metaArgs by the mergeArgs function, meaning that any other arguments -- must be nil. --]] if type(key) == 'string' then key = options.translate[key] end local val = metaArgs[key] if val ~= nil then return val elseif metatable.donePairs or nilArgs[key] then return nil end for _, argTable in ipairs(argTables) do local argTableVal = tidyVal(key, argTable[key]) if argTableVal ~= nil then metaArgs[key] = argTableVal return argTableVal end end nilArgs[key] = 'h' return nil end metatable.__newindex = function (t, key, val) -- This function is called when a module tries to add a new value to the -- args table, or tries to change an existing value. if type(key) == 'string' then key = options.translate[key] end if options.readOnly then error( 'could not write to argument table key "' .. tostring(key) .. '"; the table is read-only', 2 ) elseif options.noOverwrite and args[key] ~= nil then error( 'could not write to argument table key "' .. tostring(key) .. '"; overwriting existing arguments is not permitted', 2 ) elseif val == nil then --[[ -- If the argument is to be overwritten with nil, we need to erase -- the value in metaArgs, so that __index, __pairs and __ipairs do -- not use a previous existing value, if present; and we also need -- to memoize the nil in nilArgs, so that the value isn't looked -- up in the argument tables if it is accessed again. --]] metaArgs[key] = nil nilArgs[key] = 'h' else metaArgs[key] = val end end local function translatenext(invariant) local k, v = next(invariant.t, invariant.k) invariant.k = k if k == nil then return nil elseif type(k) ~= 'string' or not options.backtranslate then return k, v else local backtranslate = options.backtranslate[k] if backtranslate == nil then -- Skip this one. This is a tail call, so this won't cause stack overflow return translatenext(invariant) else return backtranslate, v end end end metatable.__pairs = function () -- Called when pairs is run on the args table. if not metatable.donePairs then mergeArgs(argTables) metatable.donePairs = true end return translatenext, { t = metaArgs } end local function inext(t, i) -- This uses our __index metamethod local v = t[i + 1] if v ~= nil then return i + 1, v end end metatable.__ipairs = function (t) -- Called when ipairs is run on the args table. return inext, t, 0 end return args end return arguments 3134ecce8429b810d445e29eae115e2ae4c36c53 828 26 63 2022-09-06T02:22:44Z OfficialURL 10 Created page with "-- This is a meta-module for producing message box templates, including -- {{mbox}}, {{ambox}}, {{imbox}}, {{tmbox}}, {{ombox}}, {{cmbox}} and {{fmbox}}. -- Load necessary modules. require('Module:No globals') local getArgs local yesno = require('Module:Yesno') -- Get a language object for formatDate and ucfirst. local lang = mw.language.getContentLanguage() -- Define constants local CONFIG_MODULE = 'Module:Message box/configuration' local DEMOSPACES = {talk = 'tmbox'..." Scribunto text/plain -- This is a meta-module for producing message box templates, including -- {{mbox}}, {{ambox}}, {{imbox}}, {{tmbox}}, {{ombox}}, {{cmbox}} and {{fmbox}}. -- Load necessary modules. require('Module:No globals') local getArgs local yesno = require('Module:Yesno') -- Get a language object for formatDate and ucfirst. local lang = mw.language.getContentLanguage() -- Define constants local CONFIG_MODULE = 'Module:Message box/configuration' local DEMOSPACES = {talk = 'tmbox', image = 'imbox', file = 'imbox', category = 'cmbox', article = 'ambox', main = 'ambox'} -------------------------------------------------------------------------------- -- Helper functions -------------------------------------------------------------------------------- local function getTitleObject(...) -- Get the title object, passing the function through pcall -- in case we are over the expensive function count limit. local success, title = pcall(mw.title.new, ...) if success then return title end end local function union(t1, t2) -- Returns the union of two arrays. local vals = {} for i, v in ipairs(t1) do vals[v] = true end for i, v in ipairs(t2) do vals[v] = true end local ret = {} for k in pairs(vals) do table.insert(ret, k) end table.sort(ret) return ret end local function getArgNums(args, prefix) local nums = {} for k, v in pairs(args) do local num = mw.ustring.match(tostring(k), '^' .. prefix .. '([1-9]%d*)$') if num then table.insert(nums, tonumber(num)) end end table.sort(nums) return nums end -------------------------------------------------------------------------------- -- Box class definition -------------------------------------------------------------------------------- local MessageBox = {} MessageBox.__index = MessageBox function MessageBox.new(boxType, args, cfg) args = args or {} local obj = {} -- Set the title object and the namespace. obj.title = getTitleObject(args.page) or mw.title.getCurrentTitle() -- Set the config for our box type. obj.cfg = cfg[boxType] if not obj.cfg then local ns = obj.title.namespace -- boxType is "mbox" or invalid input if args.demospace and args.demospace ~= '' then -- implement demospace parameter of mbox local demospace = string.lower(args.demospace) if DEMOSPACES[demospace] then -- use template from DEMOSPACES obj.cfg = cfg[DEMOSPACES[demospace]] elseif string.find( demospace, 'talk' ) then -- demo as a talk page obj.cfg = cfg.tmbox else -- default to ombox obj.cfg = cfg.ombox end elseif ns == 0 then obj.cfg = cfg.ambox -- main namespace elseif ns == 6 then obj.cfg = cfg.imbox -- file namespace elseif ns == 14 then obj.cfg = cfg.cmbox -- category namespace else local nsTable = mw.site.namespaces[ns] if nsTable and nsTable.isTalk then obj.cfg = cfg.tmbox -- any talk namespace else obj.cfg = cfg.ombox -- other namespaces or invalid input end end end -- Set the arguments, and remove all blank arguments except for the ones -- listed in cfg.allowBlankParams. do local newArgs = {} for k, v in pairs(args) do if v ~= '' then newArgs[k] = v end end for i, param in ipairs(obj.cfg.allowBlankParams or {}) do newArgs[param] = args[param] end obj.args = newArgs end -- Define internal data structure. obj.categories = {} obj.classes = {} -- For lazy loading of [[Module:Category handler]]. obj.hasCategories = false return setmetatable(obj, MessageBox) end function MessageBox:addCat(ns, cat, sort) if not cat then return nil end if sort then cat = string.format('[[Category:%s|%s]]', cat, sort) else cat = string.format('[[Category:%s]]', cat) end self.hasCategories = true self.categories[ns] = self.categories[ns] or {} table.insert(self.categories[ns], cat) end function MessageBox:addClass(class) if not class then return nil end table.insert(self.classes, class) end function MessageBox:setParameters() local args = self.args local cfg = self.cfg -- Get type data. self.type = args.type local typeData = cfg.types[self.type] self.invalidTypeError = cfg.showInvalidTypeError and self.type and not typeData typeData = typeData or cfg.types[cfg.default] self.typeClass = typeData.class self.typeImage = typeData.image -- Find if the box has been wrongly substituted. self.isSubstituted = cfg.substCheck and args.subst == 'SUBST' -- Find whether we are using a small message box. self.isSmall = cfg.allowSmall and ( cfg.smallParam and args.small == cfg.smallParam or not cfg.smallParam and yesno(args.small) ) -- Add attributes, classes and styles. self.id = args.id self.name = args.name if self.name then self:addClass('box-' .. string.gsub(self.name,' ','_')) end if yesno(args.plainlinks) ~= false then self:addClass('plainlinks') end for _, class in ipairs(cfg.classes or {}) do self:addClass(class) end if self.isSmall then self:addClass(cfg.smallClass or 'mbox-small') end self:addClass(self.typeClass) self:addClass(args.class) self.style = args.style self.attrs = args.attrs -- Set text style. self.textstyle = args.textstyle -- Find if we are on the template page or not. This functionality is only -- used if useCollapsibleTextFields is set, or if both cfg.templateCategory -- and cfg.templateCategoryRequireName are set. self.useCollapsibleTextFields = cfg.useCollapsibleTextFields if self.useCollapsibleTextFields or cfg.templateCategory and cfg.templateCategoryRequireName then if self.name then local templateName = mw.ustring.match( self.name, '^[tT][eE][mM][pP][lL][aA][tT][eE][%s_]*:[%s_]*(.*)$' ) or self.name templateName = 'Template:' .. templateName self.templateTitle = getTitleObject(templateName) end self.isTemplatePage = self.templateTitle and mw.title.equals(self.title, self.templateTitle) end -- Process data for collapsible text fields. At the moment these are only -- used in {{ambox}}. if self.useCollapsibleTextFields then -- Get the self.issue value. if self.isSmall and args.smalltext then self.issue = args.smalltext else local sect if args.sect == '' then sect = 'This ' .. (cfg.sectionDefault or 'page') elseif type(args.sect) == 'string' then sect = 'This ' .. args.sect end local issue = args.issue issue = type(issue) == 'string' and issue ~= '' and issue or nil local text = args.text text = type(text) == 'string' and text or nil local issues = {} table.insert(issues, sect) table.insert(issues, issue) table.insert(issues, text) self.issue = table.concat(issues, ' ') end -- Get the self.talk value. local talk = args.talk -- Show talk links on the template page or template subpages if the talk -- parameter is blank. if talk == '' and self.templateTitle and ( mw.title.equals(self.templateTitle, self.title) or self.title:isSubpageOf(self.templateTitle) ) then talk = '#' elseif talk == '' then talk = nil end if talk then -- If the talk value is a talk page, make a link to that page. Else -- assume that it's a section heading, and make a link to the talk -- page of the current page with that section heading. local talkTitle = getTitleObject(talk) local talkArgIsTalkPage = true if not talkTitle or not talkTitle.isTalkPage then talkArgIsTalkPage = false talkTitle = getTitleObject( self.title.text, mw.site.namespaces[self.title.namespace].talk.id ) end if talkTitle and talkTitle.exists then local talkText = 'Relevant discussion may be found on' if talkArgIsTalkPage then talkText = string.format( '%s [[%s|%s]].', talkText, talk, talkTitle.prefixedText ) else talkText = string.format( '%s the [[%s#%s|talk page]].', talkText, talkTitle.prefixedText, talk ) end self.talk = talkText end end -- Get other values. self.fix = args.fix ~= '' and args.fix or nil local date if args.date and args.date ~= '' then date = args.date elseif args.date == '' and self.isTemplatePage then date = lang:formatDate('F Y') end if date then self.date = string.format(" <small class='date-container'>''(<span class='date'>%s</span>)''</small>", date) end self.info = args.info if yesno(args.removalnotice) then self.removalNotice = cfg.removalNotice end end -- Set the non-collapsible text field. At the moment this is used by all box -- types other than ambox, and also by ambox when small=yes. if self.isSmall then self.text = args.smalltext or args.text else self.text = args.text end -- Set the below row. self.below = cfg.below and args.below -- General image settings. self.imageCellDiv = not self.isSmall and cfg.imageCellDiv self.imageEmptyCell = cfg.imageEmptyCell if cfg.imageEmptyCellStyle then self.imageEmptyCellStyle = 'border:none;padding:0px;width:1px' end -- Left image settings. local imageLeft = self.isSmall and args.smallimage or args.image if cfg.imageCheckBlank and imageLeft ~= 'blank' and imageLeft ~= 'none' or not cfg.imageCheckBlank and imageLeft ~= 'none' then self.imageLeft = imageLeft if not imageLeft then local imageSize = self.isSmall and (cfg.imageSmallSize or '30x30px') or '40x40px' self.imageLeft = string.format('[[File:%s|%s|link=|alt=]]', self.typeImage or 'Imbox notice.png', imageSize) end end -- Right image settings. local imageRight = self.isSmall and args.smallimageright or args.imageright if not (cfg.imageRightNone and imageRight == 'none') then self.imageRight = imageRight end end function MessageBox:setMainspaceCategories() local args = self.args local cfg = self.cfg if not cfg.allowMainspaceCategories then return nil end local nums = {} for _, prefix in ipairs{'cat', 'category', 'all'} do args[prefix .. '1'] = args[prefix] nums = union(nums, getArgNums(args, prefix)) end -- The following is roughly equivalent to the old {{Ambox/category}}. local date = args.date date = type(date) == 'string' and date local preposition = 'from' for _, num in ipairs(nums) do local mainCat = args['cat' .. tostring(num)] or args['category' .. tostring(num)] local allCat = args['all' .. tostring(num)] mainCat = type(mainCat) == 'string' and mainCat allCat = type(allCat) == 'string' and allCat if mainCat and date and date ~= '' then local catTitle = string.format('%s %s %s', mainCat, preposition, date) self:addCat(0, catTitle) catTitle = getTitleObject('Category:' .. catTitle) if not catTitle or not catTitle.exists then self:addCat(0, 'Articles with invalid date parameter in template') end elseif mainCat and (not date or date == '') then self:addCat(0, mainCat) end if allCat then self:addCat(0, allCat) end end end function MessageBox:setTemplateCategories() local args = self.args local cfg = self.cfg -- Add template categories. if cfg.templateCategory then if cfg.templateCategoryRequireName then if self.isTemplatePage then self:addCat(10, cfg.templateCategory) end elseif not self.title.isSubpage then self:addCat(10, cfg.templateCategory) end end -- Add template error categories. if cfg.templateErrorCategory then local templateErrorCategory = cfg.templateErrorCategory local templateCat, templateSort if not self.name and not self.title.isSubpage then templateCat = templateErrorCategory elseif self.isTemplatePage then local paramsToCheck = cfg.templateErrorParamsToCheck or {} local count = 0 for i, param in ipairs(paramsToCheck) do if not args[param] then count = count + 1 end end if count > 0 then templateCat = templateErrorCategory templateSort = tostring(count) end if self.categoryNums and #self.categoryNums > 0 then templateCat = templateErrorCategory templateSort = 'C' end end self:addCat(10, templateCat, templateSort) end end function MessageBox:setAllNamespaceCategories() -- Set categories for all namespaces. if self.invalidTypeError then local allSort = (self.title.namespace == 0 and 'Main:' or '') .. self.title.prefixedText self:addCat('all', 'Wikipedia message box parameter needs fixing', allSort) end if self.isSubstituted then self:addCat('all', 'Pages with incorrectly substituted templates') end end function MessageBox:setCategories() if self.title.namespace == 0 then self:setMainspaceCategories() elseif self.title.namespace == 10 then self:setTemplateCategories() end self:setAllNamespaceCategories() end function MessageBox:renderCategories() if not self.hasCategories then -- No categories added, no need to pass them to Category handler so, -- if it was invoked, it would return the empty string. -- So we shortcut and return the empty string. return "" end -- Convert category tables to strings and pass them through -- [[Module:Category handler]]. return require('Module:Category handler')._main{ main = table.concat(self.categories[0] or {}), template = table.concat(self.categories[10] or {}), all = table.concat(self.categories.all or {}), nocat = self.args.nocat, page = self.args.page } end function MessageBox:export() local root = mw.html.create() -- Add the subst check error. if self.isSubstituted and self.name then root:tag('b') :addClass('error') :wikitext(string.format( 'Template <code>%s[[Template:%s|%s]]%s</code> has been incorrectly substituted.', mw.text.nowiki('{{'), self.name, self.name, mw.text.nowiki('}}') )) end -- Create the box table. local boxTable = root:tag('table') boxTable:attr('id', self.id or nil) for i, class in ipairs(self.classes or {}) do boxTable:addClass(class or nil) end boxTable :cssText(self.style or nil) :attr('role', 'presentation') if self.attrs then boxTable:attr(self.attrs) end -- Add the left-hand image. local row = boxTable:tag('tr') if self.imageLeft then local imageLeftCell = row:tag('td'):addClass('mbox-image') if self.imageCellDiv then -- If we are using a div, redefine imageLeftCell so that the image -- is inside it. Divs use style="width: 52px;", which limits the -- image width to 52px. If any images in a div are wider than that, -- they may overlap with the text or cause other display problems. imageLeftCell = imageLeftCell:tag('div'):css('width', '52px') end imageLeftCell:wikitext(self.imageLeft or nil) elseif self.imageEmptyCell then -- Some message boxes define an empty cell if no image is specified, and -- some don't. The old template code in templates where empty cells are -- specified gives the following hint: "No image. Cell with some width -- or padding necessary for text cell to have 100% width." row:tag('td') :addClass('mbox-empty-cell') :cssText(self.imageEmptyCellStyle or nil) end -- Add the text. local textCell = row:tag('td'):addClass('mbox-text') if self.useCollapsibleTextFields then -- The message box uses advanced text parameters that allow things to be -- collapsible. At the moment, only ambox uses this. textCell:cssText(self.textstyle or nil) local textCellDiv = textCell:tag('div') textCellDiv :addClass('mbox-text-span') :wikitext(self.issue or nil) if (self.talk or self.fix) and not self.isSmall then textCellDiv:tag('span') :addClass('hide-when-compact') :wikitext(self.talk and (' ' .. self.talk) or nil) :wikitext(self.fix and (' ' .. self.fix) or nil) end textCellDiv:wikitext(self.date and (' ' .. self.date) or nil) if self.info and not self.isSmall then textCellDiv :tag('span') :addClass('hide-when-compact') :wikitext(self.info and (' ' .. self.info) or nil) end if self.removalNotice then textCellDiv:tag('small') :addClass('hide-when-compact') :tag('i') :wikitext(string.format(" (%s)", self.removalNotice)) end else -- Default text formatting - anything goes. textCell :cssText(self.textstyle or nil) :wikitext(self.text or nil) end -- Add the right-hand image. if self.imageRight then local imageRightCell = row:tag('td'):addClass('mbox-imageright') if self.imageCellDiv then -- If we are using a div, redefine imageRightCell so that the image -- is inside it. imageRightCell = imageRightCell:tag('div'):css('width', '52px') end imageRightCell :wikitext(self.imageRight or nil) end -- Add the below row. if self.below then boxTable:tag('tr') :tag('td') :attr('colspan', self.imageRight and '3' or '2') :addClass('mbox-text') :cssText(self.textstyle or nil) :wikitext(self.below or nil) end -- Add error message for invalid type parameters. if self.invalidTypeError then root:tag('div') :css('text-align', 'center') :wikitext(string.format( 'This message box is using an invalid "type=%s" parameter and needs fixing.', self.type or '' )) end -- Add categories. root:wikitext(self:renderCategories() or nil) return tostring(root) end -------------------------------------------------------------------------------- -- Exports -------------------------------------------------------------------------------- local p, mt = {}, {} function p._exportClasses() -- For testing. return { MessageBox = MessageBox } end function p.main(boxType, args, cfgTables) local box = MessageBox.new(boxType, args, cfgTables or mw.loadData(CONFIG_MODULE)) box:setParameters() box:setCategories() return box:export() end function mt.__index(t, k) return function (frame) if not getArgs then getArgs = require('Module:Arguments').getArgs end return t.main(k, getArgs(frame, {trim = false, removeBlanks = false})) end end return setmetatable(p, mt) aab916b126b6306dc8b5e6b2f81f992e2ca90342 828 27 64 2022-09-06T02:23:17Z OfficialURL 10 Created page with "local mt = getmetatable(_G) or {} function mt.__index (t, k) if k ~= 'arg' then error('Tried to read nil global ' .. tostring(k), 2) end return nil end function mt.__newindex(t, k, v) if k ~= 'arg' then error('Tried to write global ' .. tostring(k), 2) end rawset(t, k, v) end setmetatable(_G, mt)" Scribunto text/plain local mt = getmetatable(_G) or {} function mt.__index (t, k) if k ~= 'arg' then error('Tried to read nil global ' .. tostring(k), 2) end return nil end function mt.__newindex(t, k, v) if k ~= 'arg' then error('Tried to write global ' .. tostring(k), 2) end rawset(t, k, v) end setmetatable(_G, mt) 8ce3969f7d53b08bd00dabe4cc9780bc6afd412a 828 28 65 2022-09-06T02:23:49Z OfficialURL 10 Created page with "-- Function allowing for consistent treatment of boolean-like wikitext input. -- It works similarly to the template {{yesno}}. return function (val, default) -- If your wiki uses non-ascii characters for any of "yes", "no", etc., you -- should replace "val:lower()" with "mw.ustring.lower(val)" in the -- following line. val = type(val) == 'string' and val:lower() or val if val == nil then return nil elseif val == true or val == 'yes' or val == 'y' or val =..." Scribunto text/plain -- Function allowing for consistent treatment of boolean-like wikitext input. -- It works similarly to the template {{yesno}}. return function (val, default) -- If your wiki uses non-ascii characters for any of "yes", "no", etc., you -- should replace "val:lower()" with "mw.ustring.lower(val)" in the -- following line. val = type(val) == 'string' and val:lower() or val if val == nil then return nil elseif val == true or val == 'yes' or val == 'y' or val == 'true' or val == 't' or val == 'on' or tonumber(val) == 1 then return true elseif val == false or val == 'no' or val == 'n' or val == 'false' or val == 'f' or val == 'off' or tonumber(val) == 0 then return false else return default end end f767643e7d12126d020d88d662a3dd057817b9dc 828 29 66 2022-09-06T02:24:20Z OfficialURL 10 Created page with "---------------------------------------------------------------------------------------------------- -- -- Configuration for Module:Documentation -- -- Here you can set the values of the parameters and messages used in Module:Documentation to -- localise it to your wiki and your language. Unless specified otherwise, values given here -- should be string values. ---------------------------------------------------------------------------------..." Scribunto text/plain ---------------------------------------------------------------------------------------------------- -- -- Configuration for Module:Documentation -- -- Here you can set the values of the parameters and messages used in Module:Documentation to -- localise it to your wiki and your language. Unless specified otherwise, values given here -- should be string values. ---------------------------------------------------------------------------------------------------- local cfg = {} -- Do not edit this line. ---------------------------------------------------------------------------------------------------- -- Protection template configuration ---------------------------------------------------------------------------------------------------- -- cfg['protection-reason-edit'] -- The protection reason for edit-protected templates to pass to -- [[Module:Protection banner]]. cfg['protection-reason-edit'] = 'template' --[[ ---------------------------------------------------------------------------------------------------- -- Sandbox notice configuration -- -- On sandbox pages the module can display a template notifying users that the current page is a -- sandbox, and the location of test cases pages, etc. The module decides whether the page is a -- sandbox or not based on the value of cfg['sandbox-subpage']. The following settings configure the -- messages that the notices contains. ---------------------------------------------------------------------------------------------------- --]] -- cfg['sandbox-notice-image'] -- The image displayed in the sandbox notice. cfg['sandbox-notice-image'] = '[[Image:Sandbox.svg|50px|alt=|link=]]' --[[ -- cfg['sandbox-notice-pagetype-template'] -- cfg['sandbox-notice-pagetype-module'] -- cfg['sandbox-notice-pagetype-other'] -- The page type of the sandbox page. The message that is displayed depends on the current subject -- namespace. This message is used in either cfg['sandbox-notice-blurb'] or -- cfg['sandbox-notice-diff-blurb']. --]] cfg['sandbox-notice-pagetype-template'] = '[[Wikipedia:Template test cases|template sandbox]] page' cfg['sandbox-notice-pagetype-module'] = '[[Wikipedia:Template test cases|module sandbox]] page' cfg['sandbox-notice-pagetype-other'] = 'sandbox page' --[[ -- cfg['sandbox-notice-blurb'] -- cfg['sandbox-notice-diff-blurb'] -- cfg['sandbox-notice-diff-display'] -- Either cfg['sandbox-notice-blurb'] or cfg['sandbox-notice-diff-blurb'] is the opening sentence -- of the sandbox notice. The latter has a diff link, but the former does not. $1 is the page -- type, which is either cfg['sandbox-notice-pagetype-template'], -- cfg['sandbox-notice-pagetype-module'] or cfg['sandbox-notice-pagetype-other'] depending what -- namespace we are in. $2 is a link to the main template page, and $3 is a diff link between -- the sandbox and the main template. The display value of the diff link is set by -- cfg['sandbox-notice-compare-link-display']. --]] cfg['sandbox-notice-blurb'] = 'This is the $1 for $2.' cfg['sandbox-notice-diff-blurb'] = 'This is the $1 for $2 ($3).' cfg['sandbox-notice-compare-link-display'] = 'diff' --[[ -- cfg['sandbox-notice-testcases-blurb'] -- cfg['sandbox-notice-testcases-link-display'] -- cfg['sandbox-notice-testcases-run-blurb'] -- cfg['sandbox-notice-testcases-run-link-display'] -- cfg['sandbox-notice-testcases-blurb'] is a sentence notifying the user that there is a test cases page -- corresponding to this sandbox that they can edit. $1 is a link to the test cases page. -- cfg['sandbox-notice-testcases-link-display'] is the display value for that link. -- cfg['sandbox-notice-testcases-run-blurb'] is a sentence notifying the user that there is a test cases page -- corresponding to this sandbox that they can edit, along with a link to run it. $1 is a link to the test -- cases page, and $2 is a link to the page to run it. -- cfg['sandbox-notice-testcases-run-link-display'] is the display value for the link to run the test -- cases. --]] cfg['sandbox-notice-testcases-blurb'] = 'See also the companion subpage for $1.' cfg['sandbox-notice-testcases-link-display'] = 'test cases' cfg['sandbox-notice-testcases-run-blurb'] = 'See also the companion subpage for $1 ($2).' cfg['sandbox-notice-testcases-run-link-display'] = 'run' -- cfg['sandbox-category'] -- A category to add to all template sandboxes. cfg['sandbox-category'] = 'Template sandboxes' ---------------------------------------------------------------------------------------------------- -- Start box configuration ---------------------------------------------------------------------------------------------------- -- cfg['documentation-icon-wikitext'] -- The wikitext for the icon shown at the top of the template. cfg['documentation-icon-wikitext'] = '[[File:Test Template Info-Icon - Version (2).svg|50px|link=|alt=]]' -- cfg['template-namespace-heading'] -- The heading shown in the template namespace. cfg['template-namespace-heading'] = 'Template documentation' -- cfg['module-namespace-heading'] -- The heading shown in the module namespace. cfg['module-namespace-heading'] = 'Module documentation' -- cfg['file-namespace-heading'] -- The heading shown in the file namespace. cfg['file-namespace-heading'] = 'Summary' -- cfg['other-namespaces-heading'] -- The heading shown in other namespaces. cfg['other-namespaces-heading'] = 'Documentation' -- cfg['view-link-display'] -- The text to display for "view" links. cfg['view-link-display'] = 'view' -- cfg['edit-link-display'] -- The text to display for "edit" links. cfg['edit-link-display'] = 'edit' -- cfg['history-link-display'] -- The text to display for "history" links. cfg['history-link-display'] = 'history' -- cfg['purge-link-display'] -- The text to display for "purge" links. cfg['purge-link-display'] = 'purge' -- cfg['create-link-display'] -- The text to display for "create" links. cfg['create-link-display'] = 'create' ---------------------------------------------------------------------------------------------------- -- Link box (end box) configuration ---------------------------------------------------------------------------------------------------- -- cfg['transcluded-from-blurb'] -- Notice displayed when the docs are transcluded from another page. $1 is a wikilink to that page. cfg['transcluded-from-blurb'] = 'The above [[Wikipedia:Template documentation|documentation]] is [[Wikipedia:Transclusion|transcluded]] from $1.' --[[ -- cfg['create-module-doc-blurb'] -- Notice displayed in the module namespace when the documentation subpage does not exist. -- $1 is a link to create the documentation page with the preload cfg['module-preload'] and the -- display cfg['create-link-display']. --]] cfg['create-module-doc-blurb'] = 'You might want to $1 a documentation page for this [[Wikipedia:Lua|Scribunto module]].' ---------------------------------------------------------------------------------------------------- -- Experiment blurb configuration ---------------------------------------------------------------------------------------------------- --[[ -- cfg['experiment-blurb-template'] -- cfg['experiment-blurb-module'] -- The experiment blurb is the text inviting editors to experiment in sandbox and test cases pages. -- It is only shown in the template and module namespaces. With the default English settings, it -- might look like this: -- -- Editors can experiment in this template's sandbox (edit | diff) and testcases (edit) pages. -- -- In this example, "sandbox", "edit", "diff", "testcases", and "edit" would all be links. -- -- There are two versions, cfg['experiment-blurb-template'] and cfg['experiment-blurb-module'], depending -- on what namespace we are in. -- -- Parameters: -- -- $1 is a link to the sandbox page. If the sandbox exists, it is in the following format: -- -- cfg['sandbox-link-display'] (cfg['sandbox-edit-link-display'] | cfg['compare-link-display']) -- -- If the sandbox doesn't exist, it is in the format: -- -- cfg['sandbox-link-display'] (cfg['sandbox-create-link-display'] | cfg['mirror-link-display']) -- -- The link for cfg['sandbox-create-link-display'] link preloads the page with cfg['template-sandbox-preload'] -- or cfg['module-sandbox-preload'], depending on the current namespace. The link for cfg['mirror-link-display'] -- loads a default edit summary of cfg['mirror-edit-summary']. -- -- $2 is a link to the test cases page. If the test cases page exists, it is in the following format: -- -- cfg['testcases-link-display'] (cfg['testcases-edit-link-display'] | cfg['testcases-run-link-display']) -- -- If the test cases page doesn't exist, it is in the format: -- -- cfg['testcases-link-display'] (cfg['testcases-create-link-display']) -- -- If the test cases page doesn't exist, the link for cfg['testcases-create-link-display'] preloads the -- page with cfg['template-testcases-preload'] or cfg['module-testcases-preload'], depending on the current -- namespace. --]] cfg['experiment-blurb-template'] = "Editors can experiment in this template's $1 and $2 pages." cfg['experiment-blurb-module'] = "Editors can experiment in this module's $1 and $2 pages." ---------------------------------------------------------------------------------------------------- -- Sandbox link configuration ---------------------------------------------------------------------------------------------------- -- cfg['sandbox-subpage'] -- The name of the template subpage typically used for sandboxes. cfg['sandbox-subpage'] = 'sandbox' -- cfg['template-sandbox-preload'] -- Preload file for template sandbox pages. cfg['template-sandbox-preload'] = 'Template:Documentation/preload-sandbox' -- cfg['module-sandbox-preload'] -- Preload file for Lua module sandbox pages. cfg['module-sandbox-preload'] = 'Template:Documentation/preload-module-sandbox' -- cfg['sandbox-link-display'] -- The text to display for "sandbox" links. cfg['sandbox-link-display'] = 'sandbox' -- cfg['sandbox-edit-link-display'] -- The text to display for sandbox "edit" links. cfg['sandbox-edit-link-display'] = 'edit' -- cfg['sandbox-create-link-display'] -- The text to display for sandbox "create" links. cfg['sandbox-create-link-display'] = 'create' -- cfg['compare-link-display'] -- The text to display for "compare" links. cfg['compare-link-display'] = 'diff' -- cfg['mirror-edit-summary'] -- The default edit summary to use when a user clicks the "mirror" link. $1 is a wikilink to the -- template page. cfg['mirror-edit-summary'] = 'Create sandbox version of $1' -- cfg['mirror-link-display'] -- The text to display for "mirror" links. cfg['mirror-link-display'] = 'mirror' -- cfg['mirror-link-preload'] -- The page to preload when a user clicks the "mirror" link. cfg['mirror-link-preload'] = 'Template:Documentation/mirror' ---------------------------------------------------------------------------------------------------- -- Test cases link configuration ---------------------------------------------------------------------------------------------------- -- cfg['testcases-subpage'] -- The name of the template subpage typically used for test cases. cfg['testcases-subpage'] = 'testcases' -- cfg['template-testcases-preload'] -- Preload file for template test cases pages. cfg['template-testcases-preload'] = 'Template:Documentation/preload-testcases' -- cfg['module-testcases-preload'] -- Preload file for Lua module test cases pages. cfg['module-testcases-preload'] = 'Template:Documentation/preload-module-testcases' -- cfg['testcases-link-display'] -- The text to display for "testcases" links. cfg['testcases-link-display'] = 'testcases' -- cfg['testcases-edit-link-display'] -- The text to display for test cases "edit" links. cfg['testcases-edit-link-display'] = 'edit' -- cfg['testcases-run-link-display'] -- The text to display for test cases "run" links. cfg['testcases-run-link-display'] = 'run' -- cfg['testcases-create-link-display'] -- The text to display for test cases "create" links. cfg['testcases-create-link-display'] = 'create' ---------------------------------------------------------------------------------------------------- -- Add categories blurb configuration ---------------------------------------------------------------------------------------------------- --[[ -- cfg['add-categories-blurb'] -- Text to direct users to add categories to the /doc subpage. Not used if the "content" or -- "docname fed" arguments are set, as then it is not clear where to add the categories. $1 is a -- link to the /doc subpage with a display value of cfg['doc-link-display']. --]] cfg['add-categories-blurb'] = 'Please add categories to the $1 subpage.' -- cfg['doc-link-display'] -- The text to display when linking to the /doc subpage. cfg['doc-link-display'] = '/doc' ---------------------------------------------------------------------------------------------------- -- Subpages link configuration ---------------------------------------------------------------------------------------------------- --[[ -- cfg['subpages-blurb'] -- The "Subpages of this template" blurb. $1 is a link to the main template's subpages with a -- display value of cfg['subpages-link-display']. In the English version this blurb is simply -- the link followed by a period, and the link display provides the actual text. --]] cfg['subpages-blurb'] = '$1.' --[[ -- cfg['subpages-link-display'] -- The text to display for the "subpages of this page" link. $1 is cfg['template-pagetype'], -- cfg['module-pagetype'] or cfg['default-pagetype'], depending on whether the current page is in -- the template namespace, the module namespace, or another namespace. --]] cfg['subpages-link-display'] = 'Subpages of this $1' -- cfg['template-pagetype'] -- The pagetype to display for template pages. cfg['template-pagetype'] = 'template' -- cfg['module-pagetype'] -- The pagetype to display for Lua module pages. cfg['module-pagetype'] = 'module' -- cfg['default-pagetype'] -- The pagetype to display for pages other than templates or Lua modules. cfg['default-pagetype'] = 'page' ---------------------------------------------------------------------------------------------------- -- Doc link configuration ---------------------------------------------------------------------------------------------------- -- cfg['doc-subpage'] -- The name of the subpage typically used for documentation pages. cfg['doc-subpage'] = 'doc' -- cfg['file-docpage-preload'] -- Preload file for documentation page in the file namespace. cfg['file-docpage-preload'] = 'Template:Documentation/preload-filespace' -- cfg['docpage-preload'] -- Preload file for template documentation pages in all namespaces. cfg['docpage-preload'] = 'Template:Documentation/preload' -- cfg['module-preload'] -- Preload file for Lua module documentation pages. cfg['module-preload'] = 'Template:Documentation/preload-module-doc' ---------------------------------------------------------------------------------------------------- -- Print version configuration ---------------------------------------------------------------------------------------------------- -- cfg['print-subpage'] -- The name of the template subpage used for print versions. cfg['print-subpage'] = 'Print' -- cfg['print-link-display'] -- The text to display when linking to the /Print subpage. cfg['print-link-display'] = '/Print' -- cfg['print-blurb'] -- Text to display if a /Print subpage exists. $1 is a link to the subpage with a display value of cfg['print-link-display']. cfg['print-blurb'] = 'A [[Help:Books/for experts#Improving the book layout|print version]] of this template exists at $1.' .. ' If you make a change to this template, please update the print version as well.' -- cfg['display-print-category'] -- Set to true to enable output of cfg['print-category'] if a /Print subpage exists. -- This should be a boolean value (either true or false). cfg['display-print-category'] = true -- cfg['print-category'] -- Category to output if cfg['display-print-category'] is set to true, and a /Print subpage exists. cfg['print-category'] = 'Templates with print versions' ---------------------------------------------------------------------------------------------------- -- HTML and CSS configuration ---------------------------------------------------------------------------------------------------- -- cfg['main-div-id'] -- The "id" attribute of the main HTML "div" tag. cfg['main-div-id'] = 'template-documentation' -- cfg['main-div-classes'] -- The CSS classes added to the main HTML "div" tag. cfg['main-div-classes'] = 'template-documentation iezoomfix' -- cfg['start-box-linkclasses'] -- The CSS classes used for the [view][edit][history] or [create] links in the start box. cfg['start-box-linkclasses'] = 'mw-editsection-like plainlinks' -- cfg['start-box-link-id'] -- The HTML "id" attribute for the links in the start box. cfg['start-box-link-id'] = 'doc_editlinks' ---------------------------------------------------------------------------------------------------- -- {{fmbox}} template configuration ---------------------------------------------------------------------------------------------------- -- cfg['fmbox-id'] -- The id sent to the "id" parameter of the {{fmbox}} template. cfg['fmbox-id'] = 'documentation-meta-data' -- cfg['fmbox-style'] -- The value sent to the style parameter of {{fmbox}}. cfg['fmbox-style'] = 'background-color: #ecfcf4' -- cfg['fmbox-textstyle'] -- The value sent to the "textstyle parameter of {{fmbox}}. cfg['fmbox-textstyle'] = 'font-style: italic' ---------------------------------------------------------------------------------------------------- -- Tracking category configuration ---------------------------------------------------------------------------------------------------- -- cfg['display-strange-usage-category'] -- Set to true to enable output of cfg['strange-usage-category'] if the module is used on a /doc subpage -- or a /testcases subpage. This should be a boolean value (either true or false). cfg['display-strange-usage-category'] = true -- cfg['strange-usage-category'] -- Category to output if cfg['display-strange-usage-category'] is set to true and the module is used on a -- /doc subpage or a /testcases subpage. cfg['strange-usage-category'] = 'Wikipedia pages with strange ((documentation)) usage' --[[ ---------------------------------------------------------------------------------------------------- -- End configuration -- -- Don't edit anything below this line. ---------------------------------------------------------------------------------------------------- --]] return cfg 37a2dfddf613853a4d472db0179b6fe844360519 828 30 67 2022-09-06T02:24:58Z OfficialURL 10 Created page with "-------------------------------------------------------------------------------- -- Message box configuration -- -- -- -- This module contains configuration data for [[Module:Message box]]. -- -------------------------------------------------------------------------------- return { ambox = { types = { speedy = { class = 'ambox-spee..." Scribunto text/plain -------------------------------------------------------------------------------- -- Message box configuration -- -- -- -- This module contains configuration data for [[Module:Message box]]. -- -------------------------------------------------------------------------------- return { ambox = { types = { speedy = { class = 'ambox-speedy', image = 'Ambox warning pn.svg' }, delete = { class = 'ambox-delete', image = 'Ambox warning pn.svg' }, content = { class = 'ambox-content', image = 'Ambox important.svg' }, style = { class = 'ambox-style', image = 'Edit-clear.svg' }, move = { class = 'ambox-move', image = 'Merge-split-transwiki default.svg' }, protection = { class = 'ambox-protection', image = 'Semi-protection-shackle-keyhole.svg' }, notice = { class = 'ambox-notice', image = 'Information icon4.svg' } }, default = 'notice', allowBlankParams = {'talk', 'sect', 'date', 'issue', 'fix', 'subst', 'hidden'}, allowSmall = true, smallParam = 'left', smallClass = 'mbox-small-left', substCheck = true, classes = {'metadata', 'ambox'}, imageEmptyCell = true, imageCheckBlank = true, imageSmallSize = '20x20px', imageCellDiv = true, useCollapsibleTextFields = true, imageRightNone = true, sectionDefault = 'article', allowMainspaceCategories = true, templateCategory = 'Article message templates', templateCategoryRequireName = true, templateErrorCategory = 'Article message templates with missing parameters', templateErrorParamsToCheck = {'issue', 'fix', 'subst'}, removalNotice = '[[Help:Maintenance template removal|Learn how and when to remove this template message]]' }, cmbox = { types = { speedy = { class = 'cmbox-speedy', image = 'Ambox warning pn.svg' }, delete = { class = 'cmbox-delete', image = 'Ambox warning pn.svg' }, content = { class = 'cmbox-content', image = 'Ambox important.svg' }, style = { class = 'cmbox-style', image = 'Edit-clear.svg' }, move = { class = 'cmbox-move', image = 'Merge-split-transwiki default.svg' }, protection = { class = 'cmbox-protection', image = 'Semi-protection-shackle-keyhole.svg' }, notice = { class = 'cmbox-notice', image = 'Information icon4.svg' } }, default = 'notice', showInvalidTypeError = true, classes = {'cmbox'}, imageEmptyCell = true }, fmbox = { types = { warning = { class = 'fmbox-warning', image = 'Ambox warning pn.svg' }, editnotice = { class = 'fmbox-editnotice', image = 'Information icon4.svg' }, system = { class = 'fmbox-system', image = 'Information icon4.svg' } }, default = 'system', showInvalidTypeError = true, classes = {'fmbox'}, imageEmptyCell = false, imageRightNone = false }, imbox = { types = { speedy = { class = 'imbox-speedy', image = 'Ambox warning pn.svg' }, delete = { class = 'imbox-delete', image = 'Ambox warning pn.svg' }, content = { class = 'imbox-content', image = 'Ambox important.svg' }, style = { class = 'imbox-style', image = 'Edit-clear.svg' }, move = { class = 'imbox-move', image = 'Merge-split-transwiki default.svg' }, protection = { class = 'imbox-protection', image = 'Semi-protection-shackle-keyhole.svg' }, license = { class = 'imbox-license licensetpl', image = 'Imbox license.png' -- @todo We need an SVG version of this }, featured = { class = 'imbox-featured', image = 'Cscr-featured.svg' }, notice = { class = 'imbox-notice', image = 'Information icon4.svg' } }, default = 'notice', showInvalidTypeError = true, classes = {'imbox'}, imageEmptyCell = true, below = true, templateCategory = 'File message boxes' }, ombox = { types = { speedy = { class = 'ombox-speedy', image = 'Ambox warning pn.svg' }, delete = { class = 'ombox-delete', image = 'Ambox warning pn.svg' }, content = { class = 'ombox-content', image = 'Ambox important.svg' }, style = { class = 'ombox-style', image = 'Edit-clear.svg' }, move = { class = 'ombox-move', image = 'Merge-split-transwiki default.svg' }, protection = { class = 'ombox-protection', image = 'Semi-protection-shackle-keyhole.svg' }, notice = { class = 'ombox-notice', image = 'Information icon4.svg' } }, default = 'notice', showInvalidTypeError = true, classes = {'ombox'}, allowSmall = true, imageEmptyCell = true, imageRightNone = true }, tmbox = { types = { speedy = { class = 'tmbox-speedy', image = 'Ambox warning pn.svg' }, delete = { class = 'tmbox-delete', image = 'Ambox warning pn.svg' }, content = { class = 'tmbox-content', image = 'Ambox important.svg' }, style = { class = 'tmbox-style', image = 'Edit-clear.svg' }, move = { class = 'tmbox-move', image = 'Merge-split-transwiki default.svg' }, protection = { class = 'tmbox-protection', image = 'Semi-protection-shackle-keyhole.svg' }, notice = { class = 'tmbox-notice', image = 'Information icon4.svg' } }, default = 'notice', showInvalidTypeError = true, classes = {'tmbox'}, allowSmall = true, imageRightNone = true, imageEmptyCell = true, imageEmptyCellStyle = true, templateCategory = 'Talk message boxes' } } ef8171b8278c52d9c20a4149614d97cd948670c2 10 31 68 2022-09-06T02:25:46Z OfficialURL 10 Created page with "{{{{{|safesubst:}}}#invoke:String|replace|source={{{1}}}|{{{2}}}|{{{3}}}|count={{{count|}}}}}<noinclude> {{documentation}} </noinclude>" wikitext text/x-wiki {{{{{|safesubst:}}}#invoke:String|replace|source={{{1}}}|{{{2}}}|{{{3}}}|count={{{count|}}}}}<noinclude> {{documentation}} </noinclude> 8ba3925c9502aa16d1ae1887f5c7214976e6ef7f 828 32 69 2022-09-06T02:26:19Z OfficialURL 10 Created page with "--[[ This module is intended to provide access to basic string functions. Most of the functions provided here can be invoked with named parameters, unnamed parameters, or a mixture. If named parameters are used, Mediawiki will automatically remove any leading or trailing whitespace from the parameter. Depending on the intended use, it may be advantageous to either preserve or remove such whitespace. Global options ignore_errors: If set to 'true' or 1, any error c..." Scribunto text/plain --[[ This module is intended to provide access to basic string functions. Most of the functions provided here can be invoked with named parameters, unnamed parameters, or a mixture. If named parameters are used, Mediawiki will automatically remove any leading or trailing whitespace from the parameter. Depending on the intended use, it may be advantageous to either preserve or remove such whitespace. Global options ignore_errors: If set to 'true' or 1, any error condition will result in an empty string being returned rather than an error message. error_category: If an error occurs, specifies the name of a category to include with the error message. The default category is [Category:Errors reported by Module String]. no_category: If set to 'true' or 1, no category will be added if an error is generated. Unit tests for this module are available at Module:String/tests. ]] local str = {} --[[ len This function returns the length of the target string. Usage: {{#invoke:String|len|target_string|}} OR {{#invoke:String|len|s=target_string}} Parameters s: The string whose length to report If invoked using named parameters, Mediawiki will automatically remove any leading or trailing whitespace from the target string. ]] function str.len( frame ) local new_args = str._getParameters( frame.args, {'s'} ) local s = new_args['s'] or '' return mw.ustring.len( s ) end --[[ sub This function returns a substring of the target string at specified indices. Usage: {{#invoke:String|sub|target_string|start_index|end_index}} OR {{#invoke:String|sub|s=target_string|i=start_index|j=end_index}} Parameters s: The string to return a subset of i: The fist index of the substring to return, defaults to 1. j: The last index of the string to return, defaults to the last character. The first character of the string is assigned an index of 1. If either i or j is a negative value, it is interpreted the same as selecting a character by counting from the end of the string. Hence, a value of -1 is the same as selecting the last character of the string. If the requested indices are out of range for the given string, an error is reported. ]] function str.sub( frame ) local new_args = str._getParameters( frame.args, { 's', 'i', 'j' } ) local s = new_args['s'] or '' local i = tonumber( new_args['i'] ) or 1 local j = tonumber( new_args['j'] ) or -1 local len = mw.ustring.len( s ) -- Convert negatives for range checking if i < 0 then i = len + i + 1 end if j < 0 then j = len + j + 1 end if i > len or j > len or i < 1 or j < 1 then return str._error( 'String subset index out of range' ) end if j < i then return str._error( 'String subset indices out of order' ) end return mw.ustring.sub( s, i, j ) end --[[ This function implements that features of {{str sub old}} and is kept in order to maintain these older templates. ]] function str.sublength( frame ) local i = tonumber( frame.args.i ) or 0 local len = tonumber( frame.args.len ) return mw.ustring.sub( frame.args.s, i + 1, len and ( i + len ) ) end --[[ match This function returns a substring from the source string that matches a specified pattern. Usage: {{#invoke:String|match|source_string|pattern_string|start_index|match_number|plain_flag|nomatch_output}} OR {{#invoke:String|match|s=source_string|pattern=pattern_string|start=start_index |match=match_number|plain=plain_flag|nomatch=nomatch_output}} Parameters s: The string to search pattern: The pattern or string to find within the string start: The index within the source string to start the search. The first character of the string has index 1. Defaults to 1. match: In some cases it may be possible to make multiple matches on a single string. This specifies which match to return, where the first match is match= 1. If a negative number is specified then a match is returned counting from the last match. Hence match = -1 is the same as requesting the last match. Defaults to 1. plain: A flag indicating that the pattern should be understood as plain text. Defaults to false. nomatch: If no match is found, output the "nomatch" value rather than an error. If invoked using named parameters, Mediawiki will automatically remove any leading or trailing whitespace from each string. In some circumstances this is desirable, in other cases one may want to preserve the whitespace. If the match_number or start_index are out of range for the string being queried, then this function generates an error. An error is also generated if no match is found. If one adds the parameter ignore_errors=true, then the error will be suppressed and an empty string will be returned on any failure. For information on constructing Lua patterns, a form of [regular expression], see: * http://www.lua.org/manual/5.1/manual.html#5.4.1 * http://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#Patterns * http://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#Ustring_patterns ]] -- This sub-routine is exported for use in other modules function str._match( s, pattern, start, match_index, plain_flag, nomatch ) if s == '' then return str._error( 'Target string is empty' ) end if pattern == '' then return str._error( 'Pattern string is empty' ) end start = tonumber(start) or 1 if math.abs(start) < 1 or math.abs(start) > mw.ustring.len( s ) then return str._error( 'Requested start is out of range' ) end if match_index == 0 then return str._error( 'Match index is out of range' ) end if plain_flag then pattern = str._escapePattern( pattern ) end local result if match_index == 1 then -- Find first match is simple case result = mw.ustring.match( s, pattern, start ) else if start > 1 then s = mw.ustring.sub( s, start ) end local iterator = mw.ustring.gmatch(s, pattern) if match_index > 0 then -- Forward search for w in iterator do match_index = match_index - 1 if match_index == 0 then result = w break end end else -- Reverse search local result_table = {} local count = 1 for w in iterator do result_table[count] = w count = count + 1 end result = result_table[ count + match_index ] end end if result == nil then if nomatch == nil then return str._error( 'Match not found' ) else return nomatch end else return result end end -- This is the entry point for #invoke:String|match function str.match( frame ) local new_args = str._getParameters( frame.args, {'s', 'pattern', 'start', 'match', 'plain', 'nomatch'} ) local s = new_args['s'] or '' local start = tonumber( new_args['start'] ) or 1 local plain_flag = str._getBoolean( new_args['plain'] or false ) local pattern = new_args['pattern'] or '' local match_index = math.floor( tonumber(new_args['match']) or 1 ) local nomatch = new_args['nomatch'] return str._match( s, pattern, start, match_index, plain_flag, nomatch ) end --[[ pos This function returns a single character from the target string at position pos. Usage: {{#invoke:String|pos|target_string|index_value}} OR {{#invoke:String|pos|target=target_string|pos=index_value}} Parameters target: The string to search pos: The index for the character to return If invoked using named parameters, Mediawiki will automatically remove any leading or trailing whitespace from the target string. In some circumstances this is desirable, in other cases one may want to preserve the whitespace. The first character has an index value of 1. If one requests a negative value, this function will select a character by counting backwards from the end of the string. In other words pos = -1 is the same as asking for the last character. A requested value of zero, or a value greater than the length of the string returns an error. ]] function str.pos( frame ) local new_args = str._getParameters( frame.args, {'target', 'pos'} ) local target_str = new_args['target'] or '' local pos = tonumber( new_args['pos'] ) or 0 if pos == 0 or math.abs(pos) > mw.ustring.len( target_str ) then return str._error( 'String index out of range' ) end return mw.ustring.sub( target_str, pos, pos ) end --[[ str_find This function duplicates the behavior of {{str_find}}, including all of its quirks. This is provided in order to support existing templates, but is NOT RECOMMENDED for new code and templates. New code is recommended to use the "find" function instead. Returns the first index in "source" that is a match to "target". Indexing is 1-based, and the function returns -1 if the "target" string is not present in "source". Important Note: If the "target" string is empty / missing, this function returns a value of "1", which is generally unexpected behavior, and must be accounted for separatetly. ]] function str.str_find( frame ) local new_args = str._getParameters( frame.args, {'source', 'target'} ) local source_str = new_args['source'] or '' local target_str = new_args['target'] or '' if target_str == '' then return 1 end local start = mw.ustring.find( source_str, target_str, 1, true ) if start == nil then start = -1 end return start end --[[ find This function allows one to search for a target string or pattern within another string. Usage: {{#invoke:String|find|source_str|target_string|start_index|plain_flag}} OR {{#invoke:String|find|source=source_str|target=target_str|start=start_index|plain=plain_flag}} Parameters source: The string to search target: The string or pattern to find within source start: The index within the source string to start the search, defaults to 1 plain: Boolean flag indicating that target should be understood as plain text and not as a Lua style regular expression, defaults to true If invoked using named parameters, Mediawiki will automatically remove any leading or trailing whitespace from the parameter. In some circumstances this is desirable, in other cases one may want to preserve the whitespace. This function returns the first index >= "start" where "target" can be found within "source". Indices are 1-based. If "target" is not found, then this function returns 0. If either "source" or "target" are missing / empty, this function also returns 0. This function should be safe for UTF-8 strings. ]] function str.find( frame ) local new_args = str._getParameters( frame.args, {'source', 'target', 'start', 'plain' } ) local source_str = new_args['source'] or '' local pattern = new_args['target'] or '' local start_pos = tonumber(new_args['start']) or 1 local plain = new_args['plain'] or true if source_str == '' or pattern == '' then return 0 end plain = str._getBoolean( plain ) local start = mw.ustring.find( source_str, pattern, start_pos, plain ) if start == nil then start = 0 end return start end --[[ replace This function allows one to replace a target string or pattern within another string. Usage: {{#invoke:String|replace|source_str|pattern_string|replace_string|replacement_count|plain_flag}} OR {{#invoke:String|replace|source=source_string|pattern=pattern_string|replace=replace_string| count=replacement_count|plain=plain_flag}} Parameters source: The string to search pattern: The string or pattern to find within source replace: The replacement text count: The number of occurences to replace, defaults to all. plain: Boolean flag indicating that pattern should be understood as plain text and not as a Lua style regular expression, defaults to true ]] function str.replace( frame ) local new_args = str._getParameters( frame.args, {'source', 'pattern', 'replace', 'count', 'plain' } ) local source_str = new_args['source'] or '' local pattern = new_args['pattern'] or '' local replace = new_args['replace'] or '' local count = tonumber( new_args['count'] ) local plain = new_args['plain'] or true if source_str == '' or pattern == '' then return source_str end plain = str._getBoolean( plain ) if plain then pattern = str._escapePattern( pattern ) replace = mw.ustring.gsub( replace, "%%", "%%%%" ) --Only need to escape replacement sequences. end local result if count ~= nil then result = mw.ustring.gsub( source_str, pattern, replace, count ) else result = mw.ustring.gsub( source_str, pattern, replace ) end return result end --[[ simple function to pipe string.rep to templates. ]] function str.rep( frame ) local repetitions = tonumber( frame.args[2] ) if not repetitions then return str._error( 'function rep expects a number as second parameter, received "' .. ( frame.args[2] or '' ) .. '"' ) end return string.rep( frame.args[1] or '', repetitions ) end --[[ escapePattern This function escapes special characters from a Lua string pattern. See [1] for details on how patterns work. [1] https://www.mediawiki.org/wiki/Extension:Scribunto/Lua_reference_manual#Patterns Usage: {{#invoke:String|escapePattern|pattern_string}} Parameters pattern_string: The pattern string to escape. ]] function str.escapePattern( frame ) local pattern_str = frame.args[1] if not pattern_str then return str._error( 'No pattern string specified' ) end local result = str._escapePattern( pattern_str ) return result end --[[ count This function counts the number of occurrences of one string in another. ]] function str.count(frame) local args = str._getParameters(frame.args, {'source', 'pattern', 'plain'}) local source = args.source or '' local pattern = args.pattern or '' local plain = str._getBoolean(args.plain or true) if plain then pattern = str._escapePattern(pattern) end local _, count = mw.ustring.gsub(source, pattern, '') return count end --[[ endswith This function determines whether a string ends with another string. ]] function str.endswith(frame) local args = str._getParameters(frame.args, {'source', 'pattern'}) local source = args.source or '' local pattern = args.pattern or '' if pattern == '' then -- All strings end with the empty string. return "yes" end if mw.ustring.sub(source, -mw.ustring.len(pattern), -1) == pattern then return "yes" else return "" end end --[[ join Join all non empty arguments together; the first argument is the separator. Usage: {{#invoke:String|join|sep|one|two|three}} ]] function str.join(frame) local args = {} local sep for _, v in ipairs( frame.args ) do if sep then if v ~= '' then table.insert(args, v) end else sep = v end end return table.concat( args, sep or '' ) end --[[ Helper function that populates the argument list given that user may need to use a mix of named and unnamed parameters. This is relevant because named parameters are not identical to unnamed parameters due to string trimming, and when dealing with strings we sometimes want to either preserve or remove that whitespace depending on the application. ]] function str._getParameters( frame_args, arg_list ) local new_args = {} local index = 1 local value for _, arg in ipairs( arg_list ) do value = frame_args[arg] if value == nil then value = frame_args[index] index = index + 1 end new_args[arg] = value end return new_args end --[[ Helper function to handle error messages. ]] function str._error( error_str ) local frame = mw.getCurrentFrame() local error_category = frame.args.error_category or 'Errors reported by Module String' local ignore_errors = frame.args.ignore_errors or false local no_category = frame.args.no_category or false if str._getBoolean(ignore_errors) then return '' end local error_str = '<strong class="error">String Module Error: ' .. error_str .. '</strong>' if error_category ~= '' and not str._getBoolean( no_category ) then error_str = '[[Category:' .. error_category .. ']]' .. error_str end return error_str end --[[ Helper Function to interpret boolean strings ]] function str._getBoolean( boolean_str ) local boolean_value if type( boolean_str ) == 'string' then boolean_str = boolean_str:lower() if boolean_str == 'false' or boolean_str == 'no' or boolean_str == '0' or boolean_str == '' then boolean_value = false else boolean_value = true end elseif type( boolean_str ) == 'boolean' then boolean_value = boolean_str else error( 'No boolean value found' ) end return boolean_value end --[[ Helper function that escapes all pattern characters so that they will be treated as plain text. ]] function str._escapePattern( pattern_str ) return mw.ustring.gsub( pattern_str, "([%(%)%.%%%+%-%*%?%[%^%$%]])", "%%%1" ) end return str 8b43e231b9362d8aa690d115ff4245c9d1c43531 10 22 70 59 2022-09-06T02:30:41Z OfficialURL 10 wikitext text/x-wiki ==Usage== <code><nowiki>{{Wikipedia|url|name}}</nowiki></code>links to <nowiki>https://en.wikipedia.org/wiki/</nowiki>''url'' (with spaces on the URL replaced by hyphens), and references the page name as ''name''. The second argument can be excluded when the URL and the implied page name coincide (most of the time). ==Examples== *<code><nowiki>{{Wikipedia|Set}}</nowiki></code>→ {{Wikipedia|Set}} *<code><nowiki>{{Wikipedia|Ordinal number}}</nowiki></code>→ {{Wikipedia|Ordinal number}} *<code><nowiki>{{Wikipedia|Buchholz hydra#Hydra theorem|Hydra theorem}}</nowiki></code>→ {{Wikipedia|Buchholz hydra#Hydra theorem|Hydra theorem}} [[Category:External resource templates]] f096c373942a89c232a6c9cc944f66fcf961c829 10 33 71 2022-09-06T02:43:46Z OfficialURL 10 Copied from https://polytope.miraheze.org/w/index.php?title=Template:Mathworld wikitext text/x-wiki <includeonly>{{{author|Weisstein, Eric W}}}. [https://mathworld.wolfram.com/{{replace|{{#invoke:Text|ucfirstAll|{{{1|{{PAGENAME}}}}}}}| |}}.html "{{#invoke:Text|ucfirstAll|{{{2|{{{1|{{PAGENAME}}}}}}}}}}"] at MathWorld.</includeonly> <noinclude>{{documentation}}</noinclude> 9b3ea8927c485fec3da3da83c847bc34a61b2980 10 34 72 2022-09-06T02:50:37Z OfficialURL 10 Copied from https://polytope.miraheze.org/wiki/Template:Mathworld/doc wikitext text/x-wiki == Usage == <code><nowiki>{{Mathworld|url|name}}</nowiki></code>links to <nowiki>https://mathworld.wolfram.com/</nowiki>''url''<nowiki>.html</nowiki> (titlecase with with spaces removed), and references the page name as ''name''. The second argument will default to the value of the first argument when not provided. There is also an additional <code>author</code> parameter which will default to Eric W. Weissstein who is the author of the vast majority of pages on Mathworld. The bottom of every Mathworld page will indicate how to cite it. Take the author parameter from there if it is not Eric W. Weissstein. == Examples == * <code><nowiki>{{Mathworld|Polytope}}</nowiki></code>: {{Mathworld|Polytope}} * <code><nowiki>{{Mathworld|Schlaefli Symbol|Schläfli Symbol}}</nowiki></code>: {{Mathworld|Schlaefli Symbol|Schläfli Symbol}} * <code><nowiki>{{Mathworld|Affine Coordinates|author=Barile, Margherita}}</nowiki></code>: {{Mathworld|Affine coordinates|author=Barile, Margherita}} [[Category:External resource templates]] 1cfac9a21344e0347a43ed7fabd51c2b8836d841 828 35 73 2022-09-06T02:50:59Z OfficialURL 10 Created page with "local Text = { serial = "2017-11-01", suite = "Text" } --[=[ Text utilities ]=] -- local globals local PatternCJK = false local PatternCombined = false local PatternLatin = false local PatternTerminated = false local QuoteLang = false local QuoteType = false local RangesLatin = false local SeekQuote = false local function factoryQuote() -- Create quote definitions QuoteLang = { af = "bd",..." Scribunto text/plain local Text = { serial = "2017-11-01", suite = "Text" } --[=[ Text utilities ]=] -- local globals local PatternCJK = false local PatternCombined = false local PatternLatin = false local PatternTerminated = false local QuoteLang = false local QuoteType = false local RangesLatin = false local SeekQuote = false local function factoryQuote() -- Create quote definitions QuoteLang = { af = "bd", ar = "la", be = "labd", bg = "bd", ca = "la", cs = "bd", da = "bd", de = "bd", dsb = "bd", et = "bd", el = "lald", en = "ld", es = "la", eu = "la", -- fa = "la", fi = "rd", fr = "laSPC", ga = "ld", he = "ldla", hr = "bd", hsb = "bd", hu = "bd", hy = "labd", id = "rd", is = "bd", it = "ld", ja = "x300C", ka = "bd", ko = "ld", lt = "bd", lv = "bd", nl = "ld", nn = "la", no = "la", pl = "bdla", pt = "lald", ro = "bdla", ru = "labd", sk = "bd", sl = "bd", sq = "la", sr = "bx", sv = "rd", th = "ld", tr = "ld", uk = "la", zh = "ld", ["de-ch"] = "la", ["en-gb"] = "lsld", ["en-us"] = "ld", ["fr-ch"] = "la", ["it-ch"] = "la", ["pt-br"] = "ldla", ["zh-tw"] = "x300C", ["zh-cn"] = "ld" } QuoteType = { bd = { { 8222, 8220 }, { 8218, 8217 } }, bdla = { { 8222, 8220 }, { 171, 187 } }, bx = { { 8222, 8221 }, { 8218, 8217 } }, la = { { 171, 187 }, { 8249, 8250 } }, laSPC = { { 171, 187 }, { 8249, 8250 }, true }, labd = { { 171, 187 }, { 8222, 8220 } }, lald = { { 171, 187 }, { 8220, 8221 } }, ld = { { 8220, 8221 }, { 8216, 8217 } }, ldla = { { 8220, 8221 }, { 171, 187 } }, lsld = { { 8216, 8217 }, { 8220, 8221 } }, rd = { { 8221, 8221 }, { 8217, 8217 } }, x300C = { { 0x300C, 0x300D }, { 0x300E, 0x300F } } } return r end -- factoryQuote() local function fiatQuote( apply, alien, advance ) -- Quote text -- Parameter: -- apply -- string, with text -- alien -- string, with language code -- advance -- number, with level 1 or 2 local r = apply local suite if not QuoteLang then factoryQuote() end suite = QuoteLang[ alien ] if not suite then local slang = alien:match( "^(%l+)-" ) if slang then suite = QuoteLang[ slang ] end if not suite then suite = QuoteLang[ "en" ] end end if suite then local quotes = QuoteType[ suite ] if quotes then local space if quotes[ 3 ] then space = "&#160;" else space = "" end quotes = quotes[ advance ] if quotes then r = mw.ustring.format( "%s%s%s%s%s", mw.ustring.char( quotes[ 1 ] ), space, apply, space, mw.ustring.char( quotes[ 2 ] ) ) end else mw.log( "fiatQuote() " .. suite ) end end return r end -- fiatQuote() Text.char = function ( apply, again, accept ) -- Create string from codepoints -- Parameter: -- apply -- table (sequence) with numerical codepoints, or nil -- again -- number of repetitions, or nil -- accept -- true, if no error messages to be appended -- Returns: string local r if type( apply ) == "table" then local bad = { } local codes = { } local s for k, v in pairs( apply ) do s = type( v ) if s == "number" then if v < 32 and v ~= 9 and v ~= 10 then v = tostring( v ) else v = math.floor( v ) s = false end elseif s ~= "string" then v = tostring( v ) end if s then table.insert( bad, v ) else table.insert( codes, v ) end end -- for k, v if #bad == 0 then if #codes > 0 then r = mw.ustring.char( unpack( codes ) ) if again then if type( again ) == "number" then local n = math.floor( again ) if n > 1 then r = r:rep( n ) elseif n < 1 then r = "" end else s = "bad repetitions: " .. tostring( again ) end end end else s = "bad codepoints: " .. table.concat( bad, " " ) end if s and not accept then r = tostring( mw.html.create( "span" ) :addClass( "error" ) :wikitext( s ) ) end end return r or "" end -- Text.char() Text.concatParams = function ( args, apply, adapt ) -- Concat list items into one string -- Parameter: -- args -- table (sequence) with numKey=string -- apply -- string (optional); separator (default: "|") -- adapt -- string (optional); format including "%s" -- Returns: string local collect = { } for k, v in pairs( args ) do if type( k ) == "number" then v = mw.text.trim( v ) if v ~= "" then if adapt then v = mw.ustring.format( adapt, v ) end table.insert( collect, v ) end end end -- for k, v return table.concat( collect, apply or "|" ) end -- Text.concatParams() Text.containsCJK = function ( analyse ) -- Is any CJK code within? -- Parameter: -- analyse -- string -- Returns: true, if CJK detected local r if not patternCJK then patternCJK = mw.ustring.char( 91, 13312, 45, 40959, 131072, 45, 178207, 93 ) end if mw.ustring.find( analyse, patternCJK ) then r = true else r = false end return r end -- Text.containsCJK() Text.getPlain = function ( adjust ) -- Remove wikisyntax from string, except templates -- Parameter: -- adjust -- string -- Returns: string local i = adjust:find( "<!--", 1, true ) local r = adjust local j while i do j = r:find( "-->", i + 3, true ) if j then r = r:sub( 1, i ) .. r:sub( j + 3 ) else r = r:sub( 1, i ) end i = r:find( "<!--", i, true ) end -- "<!--" r = r:gsub( "(</?%l[^>]*>)", "" ) :gsub( "'''(.+)'''", "%1" ) :gsub( "''(.+)''", "%1" ) :gsub( "&nbsp;", " " ) return r end -- Text.getPlain() Text.isLatinRange = function ( adjust ) -- Are characters expected to be latin or symbols within latin texts? -- Precondition: -- adjust -- string, or nil for initialization -- Returns: true, if valid for latin only local r if not RangesLatin then RangesLatin = { { 7, 687 }, { 7531, 7578 }, { 7680, 7935 }, { 8194, 8250 } } end if not PatternLatin then local range PatternLatin = "^[" for i = 1, #RangesLatin do range = RangesLatin[ i ] PatternLatin = PatternLatin .. mw.ustring.char( range[ 1 ], 45, range[ 2 ] ) end -- for i PatternLatin = PatternLatin .. "]*$" end if adjust then if mw.ustring.match( adjust, PatternLatin ) then r = true else r = false end end return r end -- Text.isLatinRange() Text.isQuote = function ( ask ) -- Is this character any quotation mark? -- Parameter: -- ask -- string, with single character -- Returns: true, if ask is quotation mark local r if not SeekQuote then SeekQuote = mw.ustring.char( 34, -- " 39, -- ' 171, -- laquo 187, -- raquo 8216, -- lsquo 8217, -- rsquo 8218, -- sbquo 8220, -- ldquo 8221, -- rdquo 8222, -- bdquo 8249, -- lsaquo 8250, -- rsaquo 0x300C, -- CJK 0x300D, -- CJK 0x300E, -- CJK 0x300F ) -- CJK end if ask == "" then r = false elseif mw.ustring.find( SeekQuote, ask, 1, true ) then r = true else r = false end return r end -- Text.isQuote() Text.listToText = function ( args, adapt ) -- Format list items similar to mw.text.listToText() -- Parameter: -- args -- table (sequence) with numKey=string -- adapt -- string (optional); format including "%s" -- Returns: string local collect = { } for k, v in pairs( args ) do if type( k ) == "number" then v = mw.text.trim( v ) if v ~= "" then if adapt then v = mw.ustring.format( adapt, v ) end table.insert( collect, v ) end end end -- for k, v return mw.text.listToText( collect ) end -- Text.listToText() Text.quote = function ( apply, alien, advance ) -- Quote text -- Parameter: -- apply -- string, with text -- alien -- string, with language code, or nil -- advance -- number, with level 1 or 2, or nil -- Returns: quoted string local mode, slang if type( alien ) == "string" then slang = mw.text.trim( alien ):lower() else slang = mw.title.getCurrentTitle().pageLanguage if not slang then -- TODO FIXME: Introduction expected 2017-04 slang = mw.language.getContentLanguage():getCode() end end if advance == 2 then mode = 2 else mode = 1 end return fiatQuote( mw.text.trim( apply ), slang, mode ) end -- Text.quote() Text.quoteUnquoted = function ( apply, alien, advance ) -- Quote text, if not yet quoted and not empty -- Parameter: -- apply -- string, with text -- alien -- string, with language code, or nil -- advance -- number, with level 1 or 2, or nil -- Returns: string; possibly quoted local r = mw.text.trim( apply ) local s = mw.ustring.sub( r, 1, 1 ) if s ~= "" and not Text.isQuote( s, advance ) then s = mw.ustring.sub( r, -1, 1 ) if not Text.isQuote( s ) then r = Text.quote( r, alien, advance ) end end return r end -- Text.quoteUnquoted() Text.removeDiacritics = function ( adjust ) -- Remove all diacritics -- Parameter: -- adjust -- string -- Returns: string; all latin letters should be ASCII -- or basic greek or cyrillic or symbols etc. local cleanup, decomposed if not PatternCombined then PatternCombined = mw.ustring.char( 91, 0x0300, 45, 0x036F, 0x1AB0, 45, 0x1AFF, 0x1DC0, 45, 0x1DFF, 0xFE20, 45, 0xFE2F, 93 ) end decomposed = mw.ustring.toNFD( adjust ) cleanup = mw.ustring.gsub( decomposed, PatternCombined, "" ) return mw.ustring.toNFC( cleanup ) end -- Text.removeDiacritics() Text.sentenceTerminated = function ( analyse ) -- Is string terminated by dot, question or exclamation mark? -- Quotation, link termination and so on granted -- Parameter: -- analyse -- string -- Returns: true, if sentence terminated local r if not PatternTerminated then PatternTerminated = mw.ustring.char( 91, 12290, 65281, 65294, 65311 ) .. "!%.%?…][\"'%]‹›«»‘’“”]*$" end if mw.ustring.find( analyse, PatternTerminated ) then r = true else r = false end return r end -- Text.sentenceTerminated() Text.ucfirstAll = function ( adjust ) -- Capitalize all words -- Precondition: -- adjust -- string -- Returns: string with all first letters in upper case local r = " " .. adjust local i = 1 local c, j, m if adjust:find( "&" ) then r = r:gsub( "&amp;", "&#38;" ) :gsub( "&lt;", "&#60;" ) :gsub( "&gt;", "&#62;" ) :gsub( "&nbsp;", "&#160;" ) :gsub( "&thinsp;", "&#8201;" ) :gsub( "&zwnj;", "&#8204;" ) :gsub( "&zwj;", "&#8205;" ) :gsub( "&lrm;", "&#8206;" ) :gsub( "&rlm;", "&#8207;" ) m = true end while i do i = mw.ustring.find( r, "%W%l", i ) if i then j = i + 1 c = mw.ustring.upper( mw.ustring.sub( r, j, j ) ) r = string.format( "%s%s%s", mw.ustring.sub( r, 1, i ), c, mw.ustring.sub( r, i + 2 ) ) i = j end end -- while i r = r:sub( 2 ) if m then r = r:gsub( "&#38;", "&amp;" ) :gsub( "&#60;", "&lt;" ) :gsub( "&#62;", "&gt;" ) :gsub( "&#160;", "&nbsp;" ) :gsub( "&#8201;", "&thinsp;" ) :gsub( "&#8204;", "&zwnj;" ) :gsub( "&#8205;", "&zwj;" ) :gsub( "&#8206;", "&lrm;" ) :gsub( "&#8207;", "&rlm;" ) :gsub( "&#X(%x+);", "&#x%1;" ) end return r end -- Text.ucfirstAll() Text.uprightNonlatin = function ( adjust ) -- Ensure non-italics for non-latin text parts -- One single greek letter might be granted -- Precondition: -- adjust -- string -- Returns: string with non-latin parts enclosed in <span> local r Text.isLatinRange() if mw.ustring.match( adjust, PatternLatin ) then -- latin only, horizontal dashes, quotes r = adjust else local c local j = false local k = 1 local m = false local n = mw.ustring.len( adjust ) local span = "%s%s<span dir='auto' style='font-style:normal'>%s</span>" local flat = function ( a ) -- isLatin local range for i = 1, #RangesLatin do range = RangesLatin[ i ] if a >= range[ 1 ] and a <= range[ 2 ] then return true end end -- for i end -- flat() local focus = function ( a ) -- char is not ambivalent local r = ( a > 64 ) if r then r = ( a < 8192 or a > 8212 ) else r = ( a == 38 or a == 60 ) -- '&' '<' end return r end -- focus() local form = function ( a ) return string.format( span, r, mw.ustring.sub( adjust, k, j - 1 ), mw.ustring.sub( adjust, j, a ) ) end -- form() r = "" for i = 1, n do c = mw.ustring.codepoint( adjust, i, i ) if focus( c ) then if flat( c ) then if j then if m then if i == m then -- single greek letter. j = false end m = false end if j then local nx = i - 1 local s = "" for ix = nx, 1, -1 do c = mw.ustring.sub( adjust, ix, ix ) if c == " " or c == "(" then nx = nx - 1 s = c .. s else break -- for ix end end -- for ix r = form( nx ) .. s j = false k = i end end elseif not j then j = i if c >= 880 and c <= 1023 then -- single greek letter? m = i + 1 else m = false end end elseif m then m = m + 1 end end -- for i if j and ( not m or m < n ) then r = form( n ) else r = r .. mw.ustring.sub( adjust, k ) end end return r end -- Text.uprightNonlatin() Text.test = function ( about ) local r if about == "quote" then factoryQuote() r = { } r.QuoteLang = QuoteLang r.QuoteType = QuoteType end return r end -- Text.test() -- Export local p = { } function p.char( frame ) local params = frame:getParent().args local story = params[ 1 ] local codes, lenient, multiple if not story then params = frame.args story = params[ 1 ] end if story then local items = mw.text.split( story, "%s+" ) if #items > 0 then local j lenient = ( params.errors == "0" ) codes = { } multiple = tonumber( params[ "*" ] ) for k, v in pairs( items ) do if v:sub( 1, 1 ) == "x" then j = tonumber( "0" .. v ) elseif v == "" then v = false else j = tonumber( v ) end if v then table.insert( codes, j or v ) end end -- for k, v end end return Text.char( codes, multiple, lenient ) end function p.concatParams( frame ) local args local template = frame.args.template if type( template ) == "string" then template = mw.text.trim( template ) template = ( template == "1" ) end if template then args = frame:getParent().args else args = frame.args end return Text.concatParams( args, frame.args.separator, frame.args.format ) end function p.containsCJK( frame ) return Text.containsCJK( frame.args[ 1 ] or "" ) and "1" or "" end function p.getPlain( frame ) return Text.getPlain( frame.args[ 1 ] or "" ) end function p.isLatinRange( frame ) return Text.isLatinRange( frame.args[ 1 ] or "" ) and "1" or "" end function p.isQuote( frame ) return Text.isQuote( frame.args[ 1 ] or "" ) and "1" or "" end function p.listToFormat(frame) local lists = {} local pformat = frame.args["format"] local sep = frame.args["sep"] or ";" -- Parameter parsen: Listen for k, v in pairs(frame.args) do local knum = tonumber(k) if knum then lists[knum] = v end end -- Listen splitten local maxListLen = 0 for i = 1, #lists do lists[i] = mw.text.split(lists[i], sep) if #lists[i] > maxListLen then maxListLen = #lists[i] end end -- Ergebnisstring generieren local result = "" local result_line = "" for i = 1, maxListLen do result_line = pformat for j = 1, #lists do result_line = mw.ustring.gsub(result_line, "%%s", lists[j][i], 1) end result = result .. result_line end return result end function p.listToText( frame ) local args local template = frame.args.template if type( template ) == "string" then template = mw.text.trim( template ) template = ( template == "1" ) end if template then args = frame:getParent().args else args = frame.args end return Text.listToText( args, frame.args.format ) end function p.quote( frame ) local slang = frame.args[2] if type( slang ) == "string" then slang = mw.text.trim( slang ) if slang == "" then slang = false end end return Text.quote( frame.args[ 1 ] or "", slang, tonumber( frame.args[3] ) ) end function p.quoteUnquoted( frame ) local slang = frame.args[2] if type( slang ) == "string" then slang = mw.text.trim( slang ) if slang == "" then slang = false end end return Text.quoteUnquoted( frame.args[ 1 ] or "", slang, tonumber( frame.args[3] ) ) end function p.removeDiacritics( frame ) return Text.removeDiacritics( frame.args[ 1 ] or "" ) end function p.sentenceTerminated( frame ) return Text.sentenceTerminated( frame.args[ 1 ] or "" ) and "1" or "" end function p.ucfirstAll( frame ) return Text.ucfirstAll( frame.args[ 1 ] or "" ) end function p.uprightNonlatin( frame ) return Text.uprightNonlatin( frame.args[ 1 ] or "" ) end function p.zip(frame) local lists = {} local seps = {} local defaultsep = frame.args["sep"] or "" local innersep = frame.args["isep"] or "" local outersep = frame.args["osep"] or "" -- Parameter parsen for k, v in pairs(frame.args) do local knum = tonumber(k) if knum then lists[knum] = v else if string.sub(k, 1, 3) == "sep" then local sepnum = tonumber(string.sub(k, 4)) if sepnum then seps[sepnum] = v end end end end -- sofern keine expliziten Separatoren angegeben sind, den Standardseparator verwenden for i = 1, math.max(#seps, #lists) do if not seps[i] then seps[i] = defaultsep end end -- Listen splitten local maxListLen = 0 for i = 1, #lists do lists[i] = mw.text.split(lists[i], seps[i]) if #lists[i] > maxListLen then maxListLen = #lists[i] end end local result = "" for i = 1, maxListLen do if i ~= 1 then result = result .. outersep end for j = 1, #lists do if j ~= 1 then result = result .. innersep end result = result .. (lists[j][i] or "") end end return result end function p.failsafe() return Text.serial end p.Text = function () return Text end -- p.Text return p 99fbcb83d0c1b7046a886a750069a29b9938ef5a 0 36 74 2022-09-06T02:52:00Z OfficialURL 10 Created page with "A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its cardina..." wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]], whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is finite when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. == External links == * {{Wikipedia|Finite set}} * {{Mathworld|Finite Set|author=Barile, Margherita}} 92165e6a1ceba708e1e6aaa667bf09f19b2e9daf 75 74 2022-09-06T02:52:26Z OfficialURL 10 wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]], whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. == External links == * {{Wikipedia|Finite set}} * {{Mathworld|Finite Set|author=Barile, Margherita}} 2fc7e228f7023f2b4721e329fdbbae1f2911cc3f 76 75 2022-09-06T02:57:24Z OfficialURL 10 Add more info wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]], whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. Likewise, a [[cardinal]] is called '''finite''' when it's the cardinality of a finite set. Once again, finite cardinals can be identified with the natural numbers. == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of two finite ordinals is finite. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of two finite cardinals is finite. == External links == * {{Wikipedia|Finite set}} * {{Mathworld|Finite Set|author=Barile, Margherita}} 73dc6f031627772ef3d2497a251cb2e7a3a6af75 77 76 2022-09-06T02:59:05Z OfficialURL 10 wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] (a one-to-one correspondence) \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]] \(\varnothing\), whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. Likewise, a [[cardinal]] is called '''finite''' when it's the cardinality of a finite set. Once again, finite cardinals can be identified with the natural numbers. == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of two finite ordinals is finite. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of two finite cardinals is finite. == External links == * {{Wikipedia|Finite set}} * {{Mathworld|Finite Set|author=Barile, Margherita}} 8246dafb0e991337be9804ef21e8a31d3b969634 80 77 2022-09-06T03:31:42Z OfficialURL 10 wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] (a one-to-one correspondence) \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]] \(\varnothing\), whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. Likewise, a [[cardinal]] is called '''finite''' when it's the cardinality of a finite set. Once again, finite cardinals can be identified with the natural numbers. == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of two finite ordinals is finite. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of two finite cardinals is finite. == External links == * {{Mathworld|Finite Set|author=Barile, Margherita}} * {{Wikipedia|Finite set}} 76f23f6cebdf12659d549b31a153738fc09b090b 0 37 78 2022-09-06T03:16:38Z OfficialURL 10 Create article wikitext text/x-wiki A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. An [[ordinal]] is called '''infinite''' when it is the [[order type]] of an infinite [[well-ordered set]]. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite. Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if * It is '''Dedekind infinite''', that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in [https://en.wikipedia.org/wiki/Bijection bijection]. * It is in bijection with the [[disjoint union]] \(S\sqcup S\). * It is in bijection with the [[Cartesian product]] \(S\times S\). Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. Further kinds of infinite sets include [[countable]] and [[uncountable]] sets. == Properties == * Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite. * The powerset of a infinite set is infinite. * The set difference of an infinite set and a finite set is infinite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]]. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. == External links == * {{Wikipedia|Infinite set}} * {{Mathworld|Infinite Set}} 6f4d44fc048eb590d210cf2001cf774d98fff3af 79 78 2022-09-06T03:22:31Z OfficialURL 10 more info wikitext text/x-wiki A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. An [[ordinal]] is called '''infinite''' when it is the [[order type]] of an infinite [[well-ordered set]]. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite. Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if * It is '''Dedekind infinite''', that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in [https://en.wikipedia.org/wiki/Bijection bijection]. * It is in bijection with the [[disjoint union]] \(S\sqcup S\). * It is in bijection with the [[Cartesian product]] \(S\times S\). Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of the natural numbers, which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. Important kinds of infinite sets include [[countable]] and [[uncountable]] sets. == Properties == * Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite. * The powerset of a infinite set is infinite. * The set difference of an infinite set and a finite set is infinite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]]. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. == External links == * {{Wikipedia|Infinite set}} * {{Mathworld|Infinite Set}} 0716e86fdcdea55598a04c1d4c0c3c9a65e71bb5 81 79 2022-09-06T03:32:28Z OfficialURL 10 wikitext text/x-wiki A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. An [[ordinal]] is called '''infinite''' when it is the [[order type]] of an infinite [[well-ordered set]]. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite. Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if * It is '''Dedekind infinite''', that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in [https://en.wikipedia.org/wiki/Bijection bijection]. * It is in bijection with the [[disjoint union]] \(S\sqcup S\). * It is in bijection with the [[Cartesian product]] \(S\times S\). Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of the natural numbers, which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. Important kinds of infinite sets include [[countable]] and [[uncountable]] sets. == Properties == * Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite. * The powerset of a infinite set is infinite. * The set difference of an infinite set and a finite set is infinite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]]. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. == Infinity == A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts, such as denoting certain [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits] or [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an [https://en.wikipedia.org/wiki/Extended_real_number_line extended real number]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number \(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\). == External links == * {{Mathworld|Infinite Set}} * {{Mathworld|Infinity}} * {{Wikipedia|Infinite set}} * {{Wikipedia|Infinity}} 871cfe860620829d420e3a23384677351e6f88fa 82 81 2022-09-06T03:35:44Z OfficialURL 10 /* Infinity */ wikitext text/x-wiki A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. An [[ordinal]] is called '''infinite''' when it is the [[order type]] of an infinite [[well-ordered set]]. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite. Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if * It is '''Dedekind infinite''', that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in [https://en.wikipedia.org/wiki/Bijection bijection]. * It is in bijection with the [[disjoint union]] \(S\sqcup S\). * It is in bijection with the [[Cartesian product]] \(S\times S\). Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of the natural numbers, which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. Important kinds of infinite sets include [[countable]] and [[uncountable]] sets. == Properties == * Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite. * The powerset of a infinite set is infinite. * The set difference of an infinite set and a finite set is infinite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]]. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. == Infinity == A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits to infinity], [https://en.wikipedia.org/wiki/Series_(mathematics) series], and [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an object in a structure such as the [https://en.wikipedia.org/wiki/Extended_real_number_line extended real numbers]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number \(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\). == External links == * {{Mathworld|Infinite Set}} * {{Mathworld|Infinity}} * {{Wikipedia|Infinite set}} * {{Wikipedia|Infinity}} a2731d1ed4fdca3b53f20a5623952118cb3863a5 83 82 2022-09-06T03:36:23Z OfficialURL 10 wikitext text/x-wiki A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. An [[ordinal]] is called '''infinite''' when it is the [[order type]] of an infinite [[well-ordered set]]. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite. Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if * It is '''Dedekind infinite''', that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in [https://en.wikipedia.org/wiki/Bijection bijection]. * It is in bijection with the [[disjoint union]] \(S\sqcup S\). * It is in bijection with the [[Cartesian product]] \(S\times S\). Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of the natural numbers, which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. A model of this theory is provided by the [[hereditarily finite set]]s. Important kinds of infinite sets include [[countable]] and [[uncountable]] sets. == Properties == * Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite. * The powerset of a infinite set is infinite. * The set difference of an infinite set and a finite set is infinite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]]. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. == Infinity == A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits to infinity], [https://en.wikipedia.org/wiki/Series_(mathematics) series], and [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an object in a structure such as the [https://en.wikipedia.org/wiki/Extended_real_number_line extended real numbers]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number \(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\). == External links == * {{Mathworld|Infinite Set}} * {{Mathworld|Infinity}} * {{Wikipedia|Infinite set}} * {{Wikipedia|Infinity}} 72187216b6a24bfa05f6369443b0228e9fa39ffb 0 38 84 2022-09-06T03:42:58Z OfficialURL 10 Redirected page to [[Infinite]] wikitext text/x-wiki #REDIRECT [[Infinite]] b5748d590866ff6b44a59aea40df91c516ce3425 85 84 2022-09-06T03:43:30Z OfficialURL 10 Changed redirect target from [[Infinite]] to [[Infinite#Infinity]] wikitext text/x-wiki #REDIRECT [[Infinite#Infinity]] 80a65fd48611c63602c46814029903be5af8427f 0 39 86 2022-09-06T03:43:44Z OfficialURL 10 Redirected page to [[Infinite]] wikitext text/x-wiki #REDIRECT [[Infinite]] b5748d590866ff6b44a59aea40df91c516ce3425 0 40 87 2022-09-06T03:43:59Z OfficialURL 10 Redirected page to [[Finite]] wikitext text/x-wiki #REDIRECT [[Finite]] df1a43e6eaeca2fc59498c47f0ba61226495f91a 0 41 88 2022-09-06T03:44:16Z OfficialURL 10 Redirected page to [[Infinite]] wikitext text/x-wiki #REDIRECT [[Infinite]] b5748d590866ff6b44a59aea40df91c516ce3425 0 42 89 2022-09-06T03:44:23Z OfficialURL 10 Redirected page to [[Finite]] wikitext text/x-wiki #REDIRECT [[Finite]] df1a43e6eaeca2fc59498c47f0ba61226495f91a 0 43 90 2022-09-06T03:44:31Z OfficialURL 10 Redirected page to [[Infinite]] wikitext text/x-wiki #REDIRECT [[Infinite]] b5748d590866ff6b44a59aea40df91c516ce3425 0 44 91 2022-09-06T03:44:42Z OfficialURL 10 Redirected page to [[Finite]] wikitext text/x-wiki #REDIRECT [[Finite]] df1a43e6eaeca2fc59498c47f0ba61226495f91a 1 45 92 2022-09-06T05:38:06Z OfficialURL 10 Created page with "The Cantor normal form isn't a function. It's rather a representation that can be assigned to any ordinal. – ~~~~" wikitext text/x-wiki The Cantor normal form isn't a function. It's rather a representation that can be assigned to any ordinal. – [[User:OfficialURL|<span style="background-image:-webkit-linear-gradient(left,#cd0000,#686800,#007600,#007171,#4646ff,#b400b4,#cd0000);color:white;"><b>OfficialURL</b></span>]] ([[User_talk:OfficialURL|talk]]) 05:37, 6 September 2022 (UTC) 832ab79d272052ec8fb731345d93b4c008108766 93 92 2022-09-06T05:43:08Z OfficialURL 10 wikitext text/x-wiki The Cantor normal form isn't an ordinal function. It's rather a representation that can be assigned to any ordinal. I guess you could consider it as a function from ordinals to finite lists of coefficient/exponent pairs, or any other Cantor normal form representation. – [[User:OfficialURL|<span style="background-image:-webkit-linear-gradient(left,#cd0000,#686800,#007600,#007171,#4646ff,#b400b4,#cd0000);color:white;"><b>OfficialURL</b></span>]] ([[User_talk:OfficialURL|talk]]) 05:37, 6 September 2022 (UTC) 8ff4b2cedb622ca66dd02bddeacd60354c2c0636 96 93 2022-09-06T18:29:27Z Augigogigi 2 Blanked the page wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 97 96 2022-09-06T18:30:20Z Augigogigi 2 /* CNF */ new section wikitext text/x-wiki == CNF == The Cantor normal form isn't an ordinal function. It's rather a representation that can be assigned to any ordinal. I guess you could consider it as a function from ordinals to finite lists of coefficient/exponent pairs, or any other Cantor normal form representation. – [[User:OfficialURL|<span style="background-image:-webkit-linear-gradient(left,#cd0000,#686800,#007600,#007171,#4646ff,#b400b4,#cd0000);color:white;"><b>OfficialURL</b></span>]] ([[User_talk:OfficialURL|talk]]) 05:37, 6 September 2022 (UTC) → [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 18:30, 6 September 2022 (UTC) 420b436c60b98dbefed4c225485e45b7ecb66bac 98 97 2022-09-06T18:30:36Z Augigogigi 2 wikitext text/x-wiki == CNF == The Cantor normal form isn't an ordinal function. It's rather a representation that can be assigned to any ordinal. I guess you could consider it as a function from ordinals to finite lists of coefficient/exponent pairs, or any other Cantor normal form representation. – [[User:OfficialURL|<span style="background-image:-webkit-linear-gradient(left,#cd0000,#686800,#007600,#007171,#4646ff,#b400b4,#cd0000);color:white;"><b>OfficialURL</b></span>]] ([[User_talk:OfficialURL|talk]]) 05:37, 6 September 2022 (UTC) 96474da97bb44b32b2a7d319115d43d1bd623b29 99 98 2022-09-06T18:32:29Z Augigogigi 2 /* CNF */ Reply wikitext text/x-wiki == CNF == The Cantor normal form isn't an ordinal function. It's rather a representation that can be assigned to any ordinal. I guess you could consider it as a function from ordinals to finite lists of coefficient/exponent pairs, or any other Cantor normal form representation. – [[User:OfficialURL|<span style="background-image:-webkit-linear-gradient(left,#cd0000,#686800,#007600,#007171,#4646ff,#b400b4,#cd0000);color:white;"><b>OfficialURL</b></span>]] ([[User_talk:OfficialURL|talk]]) 05:37, 6 September 2022 (UTC) :this should also include notations, but [List of functions and notations] is wordy → [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 18:32, 6 September 2022 (UTC) 6aa5bcd8743e9a1828b86c75e1d4c66a1777a5fd 0 46 94 2022-09-06T06:05:30Z OfficialURL 10 Created page with "The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementio..." wikitext text/x-wiki The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). 61540961ba76535959b652c470d9842cd1526f03 95 94 2022-09-06T06:05:52Z OfficialURL 10 wikitext text/x-wiki The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). 33c1f9e44434e1ecb5ed313ca86084de6ed58a20 1 19 100 52 2022-09-06T18:33:49Z Augigogigi 2 Blanked the page wikitext text/x-wiki da39a3ee5e6b4b0d3255bfef95601890afd80709 101 100 2022-09-06T18:34:51Z Augigogigi 2 /* What the page should be */ new section wikitext text/x-wiki == What the page should be == I opened a vote in the discord as to what the page should contain, reposting here for documentation. the options: # a list of every ordinal with a page # a list of every ordinal deemed 'significant' under some definition of 'significant', with some of them not having pages # a 1000-steps-esque page with a LOT of ordinals, and bookmarks to the significant ones # something else (pls elaborate) # do C [List of Ordinals] and B [List of Significant Ordinals] awaiting results <br> → [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 18:34, 6 September 2022 (UTC) e769b09720270b2e4fd5fe8fdc8369f300a0a8de 2 47 102 2022-09-06T22:01:16Z Musi 12 bassoon wikitext text/x-wiki Hi, I'm a person who made bad notations a while ago and subsequently disowned them. I'm mainly just here so I can keep track of what's going on. 82bfed86af99ad86d7daf154d4c600a56514acff 0 17 103 46 2022-09-08T01:47:42Z C7X 9 Some ordinals wikitext text/x-wiki ==== Countable ordinals ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least recursively * The least \( (^+) \)-stable ordinal<ref name=":0" />^(p.4) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] * MORE STUFF GOES HERE^{(sort out)} ==== Uncountable ordinals ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] 6708017a295a54c19c4bc312e42e5526576a3e99 104 103 2022-09-08T02:01:50Z C7X 9 More wikitext text/x-wiki ==== Countable ordinals ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] * MORE STUFF GOES HERE^{(sort out)} ==== Uncountable ordinals ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] 048ac8f6f7c2ecb438d6c5af0f0fae0420a573a4 105 104 2022-09-08T02:11:49Z C7X 9 wikitext text/x-wiki ==== Countable ordinals ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] * MORE STUFF GOES HERE^{(sort out)} ==== Uncountable ordinals ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] 3cbd3a5277783179d31f74e0067ccdf1ce0b66ce 106 105 2022-09-08T02:12:32Z C7X 9 Most common notation instead of xkcdforums notation wikitext text/x-wiki ==== Countable ordinals ==== * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]] * MORE STUFF GOES HERE^{(sort out)} ==== Uncountable ordinals ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] cd779546e639f78aeedb0a8a14c6a643526100dd 107 106 2022-09-08T02:50:18Z C7X 9 More ordinals wikitext text/x-wiki ==== Countable ordinals ==== In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) ==== Uncountable ordinals ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] 93e435aff496219819d555879076e54f477e9397 108 107 2022-09-09T08:45:39Z Augigogigi 2 added a references section wikitext text/x-wiki ==== Countable ordinals ==== In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) ==== Uncountable ordinals ==== * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == f1eb81be0ee0ee7b0d79d8b1db672230bf2b894c 109 108 2022-09-09T08:46:36Z Augigogigi 2 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTIONS__ 7abec9b3e31c20f5c7372735aa793410cb80afed 110 109 2022-09-09T08:46:53Z Augigogigi 2 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ b41f9a7cb999b3ab17bc9deab36610ae6eee5368 111 110 2022-09-12T06:15:43Z 27.252.117.18 0 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 3906cb3c5567b84f58bca3dfaaf60a3268dadb66 112 111 2022-09-12T06:16:49Z 27.252.117.18 0 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ f49ac00f7a7e0a7bf75e4dcf17039aa88496ae08 0 48 113 2022-09-13T01:49:37Z OfficialURL 10 Created page with "An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZFC]], since [[Burali-Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with a large enough ordinal,..." wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZFC]], since [[Burali-Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is usually enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 1dc631ec06520179ff278b2a2a8bb0b9c68a479c 114 113 2022-09-13T02:38:02Z OfficialURL 10 wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZFC]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 33485148b353f115ba66a8eeef1ff500a3207fe3 0 16 115 33 2022-09-13T02:38:52Z OfficialURL 10 wikitext text/x-wiki A normal function is an [[ordinal function]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]. 9aa900c0c36ead566be392d037ffaa4f284f84ee 0 48 116 114 2022-09-13T02:39:01Z OfficialURL 10 wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 308ed359b01a42e42cf428d988e66b3365abd9b4 132 116 2022-11-14T23:58:28Z C7X 9 Some context wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]] or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. a811b23d6584825cc59b23336b1c27b2b82acc7c 133 132 2022-11-15T01:15:50Z C7X 9 Example of Veblen function being defined up to uncountable ordinal wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]]<ref>D. Probst, <nowiki>[https://boris.unibe.ch/108693/1/pro17.pdf#page=153 A modular ordinal analysis of metapredicative subsystems of second-order arithmetic]</nowiki> (2017), p.153</ref> or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 1b229d85a0006d93040dd9d3723f572d2268f113 0 49 117 2022-09-13T02:43:15Z OfficialURL 10 Created page with "The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting [[well-foundedness]] (or more directly the [[axiom of regularity]])." wikitext text/x-wiki The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting [[well-foundedness]] (or more directly the [[axiom of regularity]]). 450afcc3d9f0326a8280c7e1c8030507af0c7c66 0 17 118 112 2022-09-16T19:34:39Z EricABQ 5 added psi_0(W_W) wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ bb56e114913cc5213d7bd8d9472e6039b0d936d9 119 118 2022-09-19T19:31:21Z EricABQ 5 clarified the OCFs used wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} (this and the following \( \psi \) expressions are in [[Buchholz's function]], until specified) * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal (this and the following \( \psi \) expressions are in [[Extended Buchholz's function]], until specified) * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 77f23b329c705f342dc3257984c481626bcc9350 122 119 2022-09-19T19:43:02Z EricABQ 5 oh wait, it was already mentioned wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] * \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 070dee9268ee97a115a9d0461af671c146843bde 124 122 2022-09-19T19:46:17Z EricABQ 5 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ e367dbddb3bd09a37ed547c3789abd9f8ad6d496 125 124 2022-09-19T19:47:16Z EricABQ 5 "recursive ordinals" are ordinals less than w1ck, so this was probably meant to be "recursively large" wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVELY LARGE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ fd7d9aba32eab924e2c3b83697fae2fb10874e78 126 125 2022-09-19T19:47:44Z EricABQ 5 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVELY LARGE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 09c43b722abea47af41b6338830dfb3ae42daa3b 137 126 2023-01-12T02:33:25Z C7X 9 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVELY LARGE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * Welch's \(E_0\)-ordinals <ref>P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ d12a386f0182efabb5c30ba52a8f376e1527cb4f 138 137 2023-01-12T02:34:55Z C7X 9 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]] * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \text{Z}_{2} \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] * RECURSIVELY LARGE ORDINALS GO HERE^{(sort out)} * The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least nonprojectible ordinal<ref name=":0" />^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals ** \( \gamma \), the supremum of all clockable ordinals ** \( \zeta \), the supremum of all eventually writable ordinals ** \( \Sigma \), the supremum of all accidentally writable ordinals * Welch's \(E_0\)-ordinals <ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 0cb864db51101f7b437a7201859f5931a3f97f4c 0 50 120 2022-09-19T19:36:29Z EricABQ 5 Created page with "'''Cantor normal form''' is a standard form of writing ordinals. Every ordinal \( \alpha \) can be written uniquely as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer. When \( \alpha \) is smaller than \( \varepsilon_0 \), the exponents \( \beta_1 \) through \( \beta_k \) are all smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form a notatio..." wikitext text/x-wiki '''Cantor normal form''' is a standard form of writing ordinals. Every ordinal \( \alpha \) can be written uniquely as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer. When \( \alpha \) is smaller than \( \varepsilon_0 \), the exponents \( \beta_1 \) through \( \beta_k \) are all smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form a notation for ordinals less than \( \varepsilon_0 \). 193d875664d7251b8c50f853d8eea4d2cc3718b3 144 120 2023-03-02T20:39:58Z C7X 9 wikitext text/x-wiki '''Cantor normal form''' is a standard form of writing ordinals. Cantor's normal form theorem states that every ordinal \( \alpha \) can be written uniquely as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer. When \( \alpha \) is smaller than \( \varepsilon_0 \), the exponents \( \beta_1 \) through \( \beta_k \) are all smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form a notation for ordinals less than \( \varepsilon_0 \). 98c9e91143667f64fe9fb1ee67fe0ec8f4513240 158 144 2023-07-11T03:21:42Z Yto 4 added some links wikitext text/x-wiki '''Cantor normal form''' is a standard form of writing ordinals. Cantor's normal form theorem states that every ordinal \( \alpha \) can be written uniquely as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer. When \( \alpha \) is smaller than [[Epsilon numbers|\( \varepsilon_0 \)]], the exponents \( \beta_1 \) through \( \beta_k \) are all strictly smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form an [[ordinal notation system]] for ordinals less than \( \varepsilon_0 \). 461afb019f81bc0e7cc2f66423f9163cc1304f5f 0 51 121 2022-09-19T19:41:42Z EricABQ 5 Created page with "In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. ==Von Neumann definition== The Von Neumann definition of ordinals defines ordinals as objects in [[ZFC]]. Each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \)." wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. ==Von Neumann definition== The Von Neumann definition of ordinals defines ordinals as objects in [[ZFC]]. Each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). 44aad747229d53a0f1b4effeb5506ab75f1e0ab1 127 121 2022-10-16T00:35:10Z C7X 9 ZFC not only setting of pure sets. There are also theories with urelements /* Von Neumann definition */ wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). b5dc6c81455c8319e04b3837ff48f2b677dfbbcb 128 127 2022-10-16T01:09:50Z C7X 9 Foundations other than set theory wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. e1bb02cb99658efb5df74f251f7d536481cc4e2f 0 52 123 2022-09-19T19:44:14Z EricABQ 5 Created page with "The ordinal \( \omega^2 \) is the least ordinal which is a limit of limit ordinals. It is also the growth rate of [[Conway chained arrows]] in the fast-growing hierarchy." wikitext text/x-wiki The ordinal \( \omega^2 \) is the least ordinal which is a limit of limit ordinals. It is also the growth rate of [[Conway chained arrows]] in the fast-growing hierarchy. ba309d9301addd6a0df9783b3b695851c60dfa4c 0 53 129 2022-10-16T02:53:09Z C7X 9 Created page with "'''Fodor's lemma''' (or the '''pressing-down lemma''') is a lemma proven by Géza Fodor in 1956. The lemma states that when \(\kappa\) is an uncountable regular cardinal and \(S\) is a stationary set of ordinals \(<\kappa\), any regressive function \(f:S\to\{<\kappa\}\) must be constant on a stationary set of ordinals \(<\kappa\). ==Importance to apierology== Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundame..." wikitext text/x-wiki '''Fodor's lemma''' (or the '''pressing-down lemma''') is a lemma proven by Géza Fodor in 1956. The lemma states that when \(\kappa\) is an uncountable regular cardinal and \(S\) is a stationary set of ordinals \(<\kappa\), any regressive function \(f:S\to\{<\kappa\}\) must be constant on a stationary set of ordinals \(<\kappa\). ==Importance to apierology== Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundamental sequence system that both assigns a sequence to all countable ordinals, and has the Bachmann property. Fodor's lemma may fail without choice<ref>A. Karagila, [https://arxiv.org/abs/1610.03985 Fodor's Lemma can Fail Everywhere]</math>, but instead we can use a weakened version known as '''Neumer's theorem''' to prove this result, in which the set \(S\) is removed and replaced with \(\{<\kappa\}\) in all cases. ==References== * E. Tachtsis, [https://www.ams.org/journals/proc/2020-148-03/S0002-9939-2019-14794-8/S0002-9939-2019-14794-8.pdf Juh&aacute;sz's topological generalization of Neumer's theorem may fail in ZF] (2019). Corollary 2.7. * Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. {{reflist}} 5da1e37f601a70ee40460fd6e93c4bc14e984a9b 130 129 2022-10-16T02:55:34Z C7X 9 wikitext text/x-wiki '''Fodor's lemma''' (or the '''pressing-down lemma''') is a lemma proven by Géza Fodor in 1956. The lemma states that when \(\kappa\) is an uncountable regular cardinal and \(S\) is a stationary set of ordinals \(<\kappa\), any regressive function \(f:S\to\{<\kappa\}\) must be constant on a stationary set of ordinals \(<\kappa\). ==Importance to apierology== Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundamental sequence system that both assigns a sequence to all countable ordinals, and has the Bachmann property. Fodor's lemma may fail without choice<nowiki><ref>A. Karagila, </nowiki>[https://arxiv.org/abs/1610.03985 Fodor's Lemma can Fail Everywhere]<nowiki></ref></nowiki>, but instead we can use a weakened version known as '''Neumer's theorem''' to prove this result, in which the set \(S\) is removed and replaced with \(\{<\kappa\}\) in all cases. ==References== * E. Tachtsis, [https://www.ams.org/journals/proc/2020-148-03/S0002-9939-2019-14794-8/S0002-9939-2019-14794-8.pdf Juh&aacute;sz's topological generalization of Neumer's theorem may fail in ZF] (2019). Corollary 2.7. * Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. {{reflist}} fb10000c8564f1546dbc86686af5e0197039faed 0 1 131 30 2022-10-24T07:30:58Z 79.68.22.60 0 wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[http://cantorsattic.info/Cantor%27s_Attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> 6f2af8233d2e7f156cb35224f25a40ea79a9c459 0 54 134 2022-11-29T02:33:05Z C7X 9 Created page with "Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==Historical background== In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\), Bachmann's \(\varphi\) had a complicated definition Possible source..." wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==Historical background== In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\), Bachmann's \(\varphi\) had a complicated definition Possible sources for this section: * M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis] * W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal] * About Bridge's work on Feferman theta: W. Buchholz, Relating ordinals to proofs in a perspicuous way * About the conception of Feferman theta, and some more historical detail on pre-Feferman OCFs: S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.] f33ae279d58b94b76c30b02c0fbe34e72ca4c072 135 134 2022-11-29T02:33:26Z C7X 9 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==Historical background== In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\){{citation needed}}, Bachmann's \(\varphi\) had a complicated definition Possible sources for this section: * M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis] * W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal] * About Bridge's work on Feferman theta: W. Buchholz, Relating ordinals to proofs in a perspicuous way * About the conception of Feferman theta, and some more historical detail on pre-Feferman OCFs: S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.] 8f44279665f3248c18980ca49906ce20a45de0a2 2 2 136 28 2022-12-28T17:54:01Z Augigogigi 2 wiki is mega inactive moment wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. <span style="color:red;">'''''<big><big><big>BUY SBOCF CHEAP ONLY $1799.99</big></big></big>'''''</span> apeirolgolgdsf == My Stuff == [[User:Augigogigi/TAN|TAN]] [[User:Augigogigi/TGRx|TGRx]] [[User:Augigogigi/Mahlo_Notation|Mahlo Notation]] [[User:Augigogigi/SbOCF|SbOCF]] 1825eb332590cf638b0629a82953ba614a17a004 0 18 139 48 2023-03-02T16:47:13Z 82.8.204.174 0 wikitext text/x-wiki * [[cantor_normal_form|Cantor normal form]], or CNF * The [[veblen_hierarchy|Veblen hierarchy]] * [[buchholz_psi|Buchholz's psi]] * [[extended_buchholz_psi|Extended Buchholz's psi]] * Rathjen's OCFs^{(sort out)} * Arai's OCFs^{(sort out)} * Stegert's OCFs^{(sort out)} 8c7a1e1ad6880be69d29278e793da76810be233b 143 139 2023-03-02T17:00:19Z 82.8.204.174 0 wikitext text/x-wiki * [[cantor_normal_form|Cantor normal form]], or CNF * The [[veblen_hierarchy|Veblen hierarchy]] * [[buchholz_psi|Buchholz's psi]] * [[extended_buchholz_psi|Extended Buchholz's psi]] * Rathjen's OCFs: Michael Rathjen made a variety of ordinal collapsing functions for proof-theoretic purposes, these include: ** Rathjen's \( \psi \) for an ordinal analysis of KPM. ** Rathjen's \( \Psi \) for an ordinal analysis of KP with the \( \Pi_3 \)-reflection schema adjoined. ** Rathjen's \( \Psi \) for an ordinal analysis of lightface (parameterless) \( \Pi^1_2 \)-comprehension, which is equivalent to KP plus the assertion that there exists a stable ordinal. * Arai's OCFs^{(sort out)} * Stegert's OCFs^{(sort out)} fe5477a4c2644afe490743809bb233b591c3a473 0 55 140 2023-03-02T16:53:01Z DaveRainbowin 18 made the page for this wikitext text/x-wiki Ω is the first uncountable infinity. f73e2a166561be8bb4a49b22e187fc30ac22f9c3 146 140 2023-03-02T22:47:09Z EricABQ 5 wikitext text/x-wiki \( \omega_1 \), sometimes called \( \Omega \), is the least uncountable ordinal. 50a742beea7ed60e7149cf8a01c8efacdce999ff 0 56 141 2023-03-02T16:56:49Z 82.8.204.174 0 Created page with "The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarro..." wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF if \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \). Ordinals beyond \( \Gamma_0 \) can either be written using a variadic extension of the Veblen hierarchy, or using ordinal collapsing functions. 0551b4ece12eceb190c234dd55312cd242e61aa9 142 141 2023-03-02T16:57:19Z 82.8.204.174 0 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \). Ordinals beyond \( \Gamma_0 \) can either be written using a variadic extension of the Veblen hierarchy, or using ordinal collapsing functions. 37c279f17d4d81d829e6f8640a492fc58c6441b2 0 57 145 2023-03-02T20:42:29Z C7X 9 Created page with "A set \(M\) is admissible if \((M,\in)\) is a model of Kripke-Platek set theory. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\).<ref>Probably in Barwise somewhere</ref>" wikitext text/x-wiki A set \(M\) is admissible if \((M,\in)\) is a model of Kripke-Platek set theory. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\).<ref>Probably in Barwise somewhere</ref> 9375c0c868e3facac9b995fad37a9c07c5ecac2e 2 58 147 2023-03-06T17:19:34Z SylvieTheScientist 20 Created page with "I'm "J" on GS and Binary198 on GWiki." wikitext text/x-wiki I'm "J" on GS and Binary198 on GWiki. 60a42b2966ec3ac72f925bda3910667cf7591080 0 15 148 31 2023-07-10T21:53:02Z Yto 4 made the page nontrivial wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m0 be maximal such that the m0-th element of C has a parent if such an m0 exists, otherwise m0 is undefined. Let G and B0 be arrays such that A=G+B0+(C), where + is concatenation, and the first column in B0 contains the parent of the m0-th element of C if m0 is defined, otherwise B0 is empty. - Say that an entry in B0 "ascends" if it is in the first column of B0 or has an ancestor in the first column of B0. Define B1,B2,...,Bn as copies of B0, but in each Bi, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B0 in the same row as x. - A[n]=G+B0+B1+...+Bn, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now the proof is finished.<ref>Source will be added as soon as it's public, which should be approximately 3 hours after this edit.</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. ed9e0b09d44fbaa5ce250bc9c9e9c96ef648f3dc 149 148 2023-07-10T22:00:07Z Yto 4 subscripts are now actually subscripts. hopefully all of them. wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m0 exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now the proof is finished.<ref>Source will be added as soon as it's public, which should be approximately 3 hours after this edit.</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. 059dad77d323e2884fd84be123a97976ef8d6106 150 149 2023-07-10T22:00:52Z Yto 4 i managed to miss a subscript wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m₀ exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now the proof is finished.<ref>Source will be added as soon as it's public, which should be approximately 3 hours after this edit.</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. 580db0d69f9db222bde578051b7688497736bbf1 153 150 2023-07-10T22:15:20Z Yto 4 wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m₀ exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>Source will be added as soon as it's public, which should be in approximately 2 hours.</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. 4bdca144bb1f9c32cae56f46f12b3638f8d066e9 154 153 2023-07-11T00:16:46Z Yto 4 wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m₀ exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>Source will be added as soon as it's public.</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. 23a49a00058f2ccbb87c0e1bf3b81ab9f90af425 155 154 2023-07-11T02:46:37Z Yto 4 wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m₀ exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>https://arxiv.org/abs/2307.04606</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. 28c1d4896a30b9c24dc082c64ba7da318dae5ef6 156 155 2023-07-11T02:48:20Z Yto 4 added source wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a typical [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length), and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m₀ exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>[https://arxiv.org/abs/2307.04606 Well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. 453d4b6b9f802487e4890edec2c4d802d361f6ff 161 156 2023-07-11T04:34:08Z Yto 4 fixing stuff and mentioning values wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]] with the base of the standard form being \( \{((\underbrace{0,0,...,0,0}_n),(\underbrace{1,1,...,1,1}_n)) : n\in\mathbb{N}\} \) and the expansion A[n] of an array A at a natural number n being defined in the following way: - The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x. - If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m₀ be maximal such that the m₀-th element of C has a parent if such an m₀ exists, otherwise m₀ is undefined. Let G and B₀ be arrays such that A=G+B₀+(C), where + is concatenation, and the first column in B₀ contains the parent of the m₀-th element of C if m₀ is defined, otherwise B₀ is empty. - Say that an entry in B₀ "ascends" if it is in the first column of B₀ or has an ancestor in the first column of B₀. Define B₁,B₂,...,B_n as copies of B₀, but in each B_i, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B₀ in the same row as x. - A[n]=G+B₀+B₁+...+B_n, where + is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( ((0,0,0),(1,1,1)) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> 1a70b2aea15ebd81bb5f654ca6c635e354155e81 163 161 2023-07-11T05:22:20Z Yto 4 tex wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]] with the base of the standard form being \( \{((\underbrace{0,0,...,0,0}_n),(\underbrace{1,1,...,1,1}_n)) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: - The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). - If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. - Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). - \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( ((0,0,0),(1,1,1)) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> beafdcc3c9e22c7b2fb1a0e0ecb594a3654d6036 0 59 151 2023-07-10T22:11:26Z Yto 4 Created page with "An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is..." wikitext text/x-wiki An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. Notable ordinal notation systems include: - [[Cantor's normal form]] - [[Primitive sequence system]] - [[Pair sequence system]] - ordinal notation systems associated to [[ordinal collapsing functions]] - [[Taranovsky's ordinal notations]] - [[Patterns of resemblance]] - [[Bashicu matrix system]] - [[Y sequence]] (as long as it is well-ordered) d7879b09bf3e62f872d33ec3042c9113865abf05 152 151 2023-07-10T22:12:26Z Yto 4 wikitext text/x-wiki An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. Notable ordinal notation systems include: - [[Cantor normal form]] - [[Primitive sequence system]] - [[Pair sequence system]] - ordinal notation systems associated to [[ordinal collapsing functions]] - [[Taranovsky's ordinal notations]] - [[Patterns of resemblance]] - [[Bashicu matrix system]] - [[Y sequence]] (as long as it is well-ordered) 7107aeefd572cc6770ce0007d57b327efde2e67e 2 12 157 25 2023-07-11T03:17:36Z Yto 4 Replaced content with "Hi, i've been an apeirologist for approximately 5 years. I recently published a [https://arxiv.org/abs/2307.04606 proof] that [[Bashicu matrix system | BMS]] is well-founded." wikitext text/x-wiki Hi, i've been an apeirologist for approximately 5 years. I recently published a [https://arxiv.org/abs/2307.04606 proof] that [[Bashicu matrix system | BMS]] is well-founded. feed131b5af5e3b8e47ec488252c10ea867c544e 0 60 159 2023-07-11T04:03:06Z Yto 4 Created page with "A '''sequence system''' is an [[ordinal notation system]] in which sequences are well-ordered. Typically, it is an [[expansion system]], with the expansion chosen so that x[n] is always lexicographically smaller than x, and additionally, so that x[0] is x without its last element and x[n] is always a subsequence of x[n+1]. If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicogr..." wikitext text/x-wiki A '''sequence system''' is an [[ordinal notation system]] in which sequences are well-ordered. Typically, it is an [[expansion system]], with the expansion chosen so that x[n] is always lexicographically smaller than x, and additionally, so that x[0] is x without its last element and x[n] is always a subsequence of x[n+1]. If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> Notable sequence systems include [[Primitive sequence system]], [[Pair sequence system]], [[Sudden sequence system]], [[Bashicu matrix system]] and [[Y sequence]]. 06bc63c8270854df9a4b0123039d3cd08abab6da 162 159 2023-07-11T04:35:15Z Yto 4 tex wikitext text/x-wiki A '''sequence system''' is an [[ordinal notation system]] in which sequences are well-ordered. Typically, it is an [[expansion system]], with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> Notable sequence systems include [[Primitive sequence system]], [[Pair sequence system]], [[Sudden sequence system]], [[Bashicu matrix system]] and [[Y sequence]]. 16fb262ed260e4a4d866ae62c447868e77f5e540 0 61 160 2023-07-11T04:07:47Z Yto 4 Created page with "An '''expansion system''' is an [[ordinal notation system]] defined in a special way. It is defined through expansion, with standard form constructed from a specified set called the base of the standard form (usually with order type \( \omega \)). More precisely, the definition involves only a set S of well-formed terms, a function \( []: S\times\mathbb{N}\to S \) (where [](x,n) is written as x[n]), and a set \( X_0 \). Then with \( X \) being the closure of \( X_0 \) u..." wikitext text/x-wiki An '''expansion system''' is an [[ordinal notation system]] defined in a special way. It is defined through expansion, with standard form constructed from a specified set called the base of the standard form (usually with order type \( \omega \)). More precisely, the definition involves only a set S of well-formed terms, a function \( []: S\times\mathbb{N}\to S \) (where [](x,n) is written as x[n]), and a set \( X_0 \). Then with \( X \) being the closure of \( X_0 \) under \( x\mapsto x[n] \) for every \( n\in\mathbb{N} \), and with \( x<y \) equivalent to the existence of \( m,n_0,n_1,...,n_m\in\mathbb{N} \) such that \( x=y[n_0][n_1]...[n_m] \), the expansion system is the set \( X \) ordered by \( < \). It still needs to be well-ordered in order to be an ordinal notation system, and the definition simply having this form is not enough to imply well-orderedness. Many expansion systems are also [[sequence system | sequence systems]], although there can be exceptions. a4964bdf73c9ff7c2cd4bcd72104f3af7b8a5c6d 164 160 2023-07-11T05:49:44Z Yto 4 wikitext text/x-wiki An '''expansion system''' is an [[ordinal notation system]] defined in a special way. It is defined through expansion, with standard form constructed from a specified set called the base of the standard form (usually with order type \( \omega \)). More precisely, the definition involves only a set \( S \) of well-formed terms, a function \( []: S\times\mathbb{N}\to S \) (where [](x,n) is written as x[n]), and a set \( X_0 \). Then with \( X \) being the closure of \( X_0 \) under \( x\mapsto x[n] \) for every \( n\in\mathbb{N} \), and with \( x<y \) equivalent to the existence of \( m,n_0,n_1,...,n_m\in\mathbb{N} \) such that \( x=y[n_0][n_1]...[n_m] \), the expansion system is the set \( X \) ordered by \( < \). Such an ordered set still needs to be well-ordered in order to be an ordinal notation system, and the definition simply having this form is not enough to imply well-orderedness. Expansion systems may also be called "dom-type systems" or similarly, in reference to the fact that some of them have a function \( dom \) assigning a "domain" to each element \( x\in S \), so that \( x[\alpha] \) is defined precisely for \( y\in dom(x) \). However, restricting the domain of an element is often not necessary, especially when the domain can only be \( \varnothing, \{0\} \) or \( \mathbb{N} \). Many expansion systems are also [[sequence system | sequence systems]], although there can be exceptions. d92823780868ec2fb38c4fe1997804f9491c4b78 0 62 165 2023-07-11T06:16:43Z Yto 4 Created page with "'''Primitive sequence system''' ('''PrSS''') is an [[ordinal notation system]] defined by [[BashicuHyudora]]. It is also a [[sequence system]] with sequences of natural numbers, and an [[expansion system]] with the base of standard form being \( \{(0,1,2,...,n) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: If \( B \) is the subsequence of \( S \) such that the first element of \( B \) is the last element of \( S \) strictly smaller..." wikitext text/x-wiki '''Primitive sequence system''' ('''PrSS''') is an [[ordinal notation system]] defined by [[BashicuHyudora]]. It is also a [[sequence system]] with sequences of natural numbers, and an [[expansion system]] with the base of standard form being \( \{(0,1,2,...,n) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: If \( B \) is the subsequence of \( S \) such that the first element of \( B \) is the last element of \( S \) strictly smaller than the last element of \( S \), and the last element of \( B \) is the second-to-last element of \( S \), and \( G \) is the subsequence of all elements of \( S \) before \( B \), then \( S[n]=G+B+\underbrace{B+B+...+B+B}_n \) The order type of PrSS is \( \varepsilon_0 \), and there is a perfect correspondence between PrSS and iterated [[Cantor normal form]]: If we define the parent of an element \( x \) of a sequence \( S \) as the last element of \( S \) before \( x \) that is smaller than \( x \) (making the first element of \( B \) also the parent of the last element of \( S \) in the definition of expansion), then a map from \( S \) to \( \varepsilon_0 \) can be defined, which maps each element \( x \) to \( \omega^{\alpha_0+\alpha_1+...+\alpha_n} \), where \( \alpha_0,\alpha_1,...,\alpha_n \) are the ordinals to which this maps the elements of \( S \) whose parent is \( x \), in the order in which they appear in \( S \). The order type of the set of sequences smaller than \( S \) is then the sum of ordinals to which the zeroes in \( S \) are mapped. This correspondence follows by transfinite induction. PrSS is identical to [[Pair sequence system]] with all pairs of the form \( (n,0) \), and to one-row [[Bashicu matrix system]]. f57a1c55e0526089997244e3edcc588288dc3310 0 63 166 2023-07-11T06:26:50Z Yto 4 Created page with "'''Pair sequence system''' ('''PSS''') is an [[ordinal notation system]] defined by [[BashicuHyudora]]. It is also a [[sequence system]] with sequences of pairs of natural numbers, and an [[expansion system]] with the base of standard form being \( \{((0,0),(1,1),(2,2),...,(n,n)) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: - The parent of a pair \( x \) in \( S \) is the last pair before it with a smaller first element. The ance..." wikitext text/x-wiki '''Pair sequence system''' ('''PSS''') is an [[ordinal notation system]] defined by [[BashicuHyudora]]. It is also a [[sequence system]] with sequences of pairs of natural numbers, and an [[expansion system]] with the base of standard form being \( \{((0,0),(1,1),(2,2),...,(n,n)) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: - The parent of a pair \( x \) in \( S \) is the last pair before it with a smaller first element. The ancestors of \( x \) are recursively defined as the parent of \( x \) and the ancestors of the parent of \( x \). Let \( B_0 \) be the subsequence of \( S \) such that the first pair in \( B_0 \) is the last ancestor of the last pair in \( S \) with the ancestor's second element being strictly smaller than the second element of the last pair in \( S \), and the last pair in \( B_0 \) is the second-to-last pair in \( S \). Then let \( G \) be the subsequence of all pairs in \( S \) before \( B_0 \), and let \( B_i \) be \( B_0 \) but with the first element of each pair increased by \( i \) times the difference between the first element of the last pair in \( S \) and the first element of the first pair in \( B_0 \). Then \( S[n]=G+B_0+B_1+B_2+...+B_n \). The order type of PSS is \( \psi(\Omega_\omega) \) in [[Buchholz's ordinal collapsing function]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of PSS]</ref> PSS is identical to two-row [[Bashicu matrix system]]. 05c8353d128060b68fe610f106cad6e70cd43097 0 17 167 138 2023-07-11T11:46:46Z 82.8.204.174 0 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinall = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * Welch's \(E_0\)-ordinals <ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[mahlo_cardinal|mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 84fe97fa50e5c0e103bcd6bc75283ac983afd232 174 167 2023-08-28T11:55:58Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * Welch's \(E_0\)-ordinals <ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal * \( I \), the smallest [[inaccessible_cardinal|inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly_compact_cardinal|weakly compact cardinal]] == References == __NOEDITSECTION__ 758e2348b1ef8adbb8922fefb091783115123840 176 174 2023-08-28T15:11:51Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * Welch's \(E_0\)-ordinals <ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 769c0ee5bde7b6b532ad4af8b64d629efc819ff8 186 176 2023-08-29T18:10:07Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 4b0535c93c038c98660ee4d51a1dc4b751c6ece5 187 186 2023-08-29T18:14:46Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ cdb7414153bf0972be42b6c968e72047477a8424 197 187 2023-08-29T21:33:26Z C7X 9 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * [[omega^3|\( \omega^{3} \)]] * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 9ccf6a1cf9ce7a2a15f010da071d87fe5d4391e0 0 59 168 152 2023-07-11T17:40:33Z Yto 4 wikitext text/x-wiki An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. Notable ordinal notation systems include: - [[Cantor normal form]] - [[Primitive sequence system]] - [[Pair sequence system]] - Ordinal notation systems associated to [[ordinal collapsing functions]] - [[Taranovsky's ordinal notations]] (the ones that are well-ordered) - [[Patterns of resemblance]] - [[Bashicu matrix system]] - [[Y sequence]] (as long as it is well-ordered) 0274d1babe2b429e7e617a47e2eea0d3fa5f4a36 169 168 2023-07-11T19:01:45Z Yto 4 wikitext text/x-wiki An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences of other such objects, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. Notable ordinal notation systems include: - [[Cantor normal form]] - [[Primitive sequence system]] - [[Pair sequence system]] - Ordinal notation systems associated to [[ordinal collapsing functions]] - [[Taranovsky's ordinal notations]] (the ones that are well-ordered) - [[Patterns of resemblance]] - [[Bashicu matrix system]] - [[Y sequence]] (as long as it is well-ordered) As can be seen in this list, proposed ordinal notation systems need to be [[Proving well-orderedness | proven]] well-ordered in order to be considered ordinal notation systems with certainty, and there are notable cases where this is not proven yet. d688e7f18ffcf42b14b08b12b534cd6445d8a864 0 64 170 2023-07-11T20:20:38Z Yto 4 Created page with "Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitraril..." wikitext text/x-wiki Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitrarily obscure cases, of course, the proof can be arbitrarily unusual itself. However, for common special cases, there are some shared features of the proof. <h2>[[Expansion system | Expansion systems]]</h2> In an expansion system, totality translates to "for every pair of terms \( x,y \), one is reachable from the other by expansion". The order can usually easily be proven to respect a lexicographical order or some other order known to be total (i.e. one can prove \( x\prec y \) implies \( x<y \), where \( \prec \) is the order of the expansion system and \( < \) is a total order), and then it is simply about proving that, intuitively, repeated expansion is never forced to skip any specific term. <br>Then a common property that directly leads to totality is the conjunction of \( x[n]\preceq x[n+1] \), \( x[n]\prec y\preceq x[n+1]\Rightarrow x[n]\preceq y[0] \), and the statement that iterating \( [0] \) always reaches the minimum eventually, no matter what you start with. The intuitive reason why this implies totality is that if we have \( x<y\prec z \) and \( x=z[n_0][n_1]...[n_m] \), then \( x \) can be reached from \( y \) by repeating \( [0] \) until it reaches something of the form \( z[n_0][n_1]...[n_k][a] \) with \( a>n_{k+1} \), at which point \( [b] \) is used with \( b \) minimal so that this doesn't go below \( x \), and the whole process repeats. The fact that this eventually terminates follows from looking at the \( (n_0,n_1,...,n_k,a) \) that appear that way, and noticing that this decreases lexicographically as the process moves on the sequence always decreases lexicographically and its length is bounded by m+1, so this form can only be reached finitely many times, and between all that, we're only iterating \( [0] \), which is guaranteed to decrease the term as much as we need, eventually getting us to \( x \) only by expanding \( y \). Keep in mind that this is not a formal proof. <br>It is not always true that this property holds. This page is currently unfinished. 35896cf4be7eaee29c726d1119ec22dcef0fa619 171 170 2023-07-11T20:25:44Z Yto 4 wikitext text/x-wiki Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitrarily obscure cases, of course, the proof can be arbitrarily unusual itself. However, for common special cases, there are some shared features of the proof. <h2>[[Expansion system | Expansion systems]]</h2> In an expansion system, totality translates to "for every pair of terms \( x,y \), one is reachable from the other by expansion". The order can usually easily be proven to respect a lexicographical order or some other order known to be total (i.e. one can prove \( x\prec y \) implies \( x<y \), where \( \prec \) is the order of the expansion system and \( < \) is a total order), and then it is simply about proving that, intuitively, repeated expansion is never forced to skip any specific term. <br>Then a common property that directly leads to totality is the conjunction of \( x[n]\preceq x[n+1] \), \( x[n]\prec y\preceq x[n+1]\Rightarrow x[n]\preceq y[0] \), and the statement that iterating \( [0] \) always reaches the minimum eventually, no matter what you start with. The intuitive reason why this implies totality is that if we have \( x<y\prec z \) and \( x=z[n_0][n_1]...[n_m] \), then \( x \) can be reached from \( y \) by repeating \( [0] \) until it reaches something of the form \( z[n_0][n_1]...[n_k][a] \) with \( a>n_{k+1} \), at which point \( [b] \) is used with \( b \) minimal so that this doesn't go below \( x \), and the whole process repeats. The fact that this eventually terminates follows from looking at the \( (n_0,n_1,...,n_k,a) \) that appear that way, and noticing that this decreases lexicographically as the process moves on the sequence always decreases lexicographically and its length is bounded by \( m+1 \), so this form can only be reached finitely many times, and between all that, we're only iterating \( [0] \), which is guaranteed to decrease the term as much as we need, eventually getting us to \( x \) only by expanding \( y \). Keep in mind that this is not a formal proof. <br>It is not always true that this property holds. This page is currently unfinished. 55136a51229ca84711bb531fead496230b69f5b7 0 15 172 163 2023-08-27T21:18:04Z Yto 4 added a bit of intuition wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{((\underbrace{0,0,...,0,0}_n),(\underbrace{1,1,...,1,1}_n)) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: - The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). - If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. - Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). - \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs). Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( ((0,0,0),(1,1,1)) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>References</h2> <references /> 0eacb2ede0712e08d2aebd07af7bb368caef3a6d 2 65 173 2023-08-28T11:45:53Z RhubarbJayde 25 Added my profile wikitext text/x-wiki I like maths, of course - main interests are among pure maths (foundations and apeirology), trigonometry & calculus. I also enjoy theoretical physics and organic chemistry. I also sometimes watch anime or play video games. Also known as: - set theory goddess (display name) on Discord - vopenka_cardinal on Discord & Youtube - BinaryCrown on Github - Binary198 on Wikipedia & GWiki - GalacticSylvie in Geometry Dash - Jayde 💜✨🏳️‍🌈 on Spotify - Jayde198 on Steam You can casually call me Jayde or Sylvie (I primarily use the former in real life). 3a471d7c3e640d5f3ed12f7f98519ae6eed09280 0 66 175 2023-08-28T12:49:04Z RhubarbJayde 25 Created page with "Large cardinals are cardinals typically defined as satisfying certain combinatorial or reflection-type properties. Their existence is asserted by various large cardinal axioms, which are usually unprovable in \( \mathrm{ZFC} \), assuming its consistency. This is because almost all large cardinals, if they exist, are worldly: a worldly cardinal is a \( \kappa \) so that \( V_\kappa \) satisfies ZFC, and thus Gödel's second incompleteness theorem applies. Due to issues..." wikitext text/x-wiki Large cardinals are cardinals typically defined as satisfying certain combinatorial or reflection-type properties. Their existence is asserted by various large cardinal axioms, which are usually unprovable in \( \mathrm{ZFC} \), assuming its consistency. This is because almost all large cardinals, if they exist, are worldly: a worldly cardinal is a \( \kappa \) so that \( V_\kappa \) satisfies ZFC, and thus Gödel's second incompleteness theorem applies. Due to issues that occur later on, such as the identity crisis and the fact that \( \Omega_\omega \) can have large cardinal properties yet not be worldly, it is typically not possible to compare large cardinals in terms of size. More common is comparing them in terms of consistency strength - an assertion \( A \) has higher consistency strength than \( B \) if \( \mathrm{ZFC} + A \) proves the consistency of \( \mathrm{ZFC} + B \). Large cardinals near the bottom of the hierarchy (both in terms of size and consistency strength) have some limited usage in ordinal collapsing functions. These are used in ordinal analysis, and convert large ordinals such as the \( \nu \)th uncountable cardinal, \( \Omega_\nu \) into countable ordinals. For analysis of theories such as \( \mathrm{KPi} \), Kripke-Platek set theory with the assertion that every set is contained in an admissible set, \( \mathrm{KPM} \), Kripke-Platek set theory with an admissible reflection principle, and beyond, large cardinals such as weakly inaccessible or weakly Mahlo cardinals are used as an additional "layer of diagonalization". However, most large cardinals, especially at the level of supercompact cardinals and beyond, likely do not have much proof-theoretic use. Instead, large cardinals are typically developed and used in the literature to measure the consistency strength of other combinatorial statements, such as resurrection axioms, quasi-projective determinacy, Chang's conjecture or the saturation of ideals. Very large cardinals, at the level of \( 0^\sharp \) and above, imply that there are nonconstructible sets, and thus can not exist within the constructible universe of sets. This has led to the study of inner model theory, in which one attempts to construct canonical models of set theory capable of accommodating large cardinals. The current state of the art is a Woodin cardinal which is a limit of Woodin cardinals, and it is difficult to continue past this point because the well-ordering of the reals in inner models for Woodin cardinals becomes harder and harder to define. There is a vast range of large cardinal axioms. Below they are listed in ascending order, in terms of consistency strength. * Worldly cardinals and the worldly hierarchy * Weakly and strongly inaccessible cardinals * \(\alpha\)-inaccessible cardinals for \( \alpha > 0 \) and the inaccessible hierarchy * Reflecting cardinals * \( \mathrm{Ord} \) is Mahlo * Uplifting cardinals * \( \Sigma_n \)-Mahlo cardinals * Weakly and strongly Mahlo cardinals * The Mahlo hierarchy, including greatly Mahlo cardinals * \( \Sigma_n \)-weakly compact cardinals * Weakly compact cardinals * \( \Pi^n_m\)-indescribable cardinals * \( \eta \)-indescribable and \( \eta \)-shrewd cardinals, for \( \eta > \omega \) * Unfoldable and strongly unfoldable cardinals * Superstrong unfoldable = strongly uplifting cardinals * Ethereal, subtle and (weakly) ineffable cardinals * The \( n \)-subtle, \( n \)-almost ineffable, \( n \)-ineffable and \( n \)-Ramsey hierarchy * Completely ineffable cardinals * Weakly Ramsey and \( \omega \)-Ramsey cardinals * Remarkable = virtually supercompact, virtually measurable and strategic \( \omega \)-Ramsey cardinals * Virtually extendible and completely remarkable cardinals * The \( n \)-iterable and the virtually \( n \)-huge* hierarchy * Virtually rank-into-rank cardinals * \( \omega \)-Erdős cardinals * The \( \alpha \)-Erdős and \( \alpha \)-iterable hierarchy * \( 0^\sharp \) * \( \omega_1 \)-iterable cardinals * \( \omega_1 \)-Erdős cardinals * Almost Ramsey cardinals * Greatly Erdős cardinals * Ramsey, Jónsson and Rowbottom cardinals * Measurable cardinal * \( 0^\dagger \) * Mitchell rank * The tall and strong hierarchies * Woodin cardinals * The axiom of (projective) determinacy * Shelah cardinals * Superstrong cardinals * Subcompact cardinals * Strongly compact cardinals * Supercompact cardinals * Woodin for strong compactness cardinals * The extendible hierarchy * Vopěnka cardinals = Woodin for supercompactness cardinals * Shelah for supercompactness cardinals * Almost high-jump and high-jump cardinals * Super high-jump and high-jump with unbounded excess closure cardinals * The huge and \( n \)-superstrong for \( n > 1 \) hierarchy * \( n \)-fold variants of large cardinals * \( \mathrm{I}^n_4 \) cardinals * Rank-into-rank cardinals and \( \omega \)-fold variants of large cardinals * The Reinhardt hierarchy * The Berkeley hierarchy 919e2d1e8e1a22df0e1576a579c51ef757a8c6a8 0 67 177 2023-08-28T15:21:23Z RhubarbJayde 25 Created page with "Countability is a key notion in set theory and apeirology. A set is called countable if it has the same size as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. Georg Cantor, the founder of set theory, proved that the set of inte..." wikitext text/x-wiki Countability is a key notion in set theory and apeirology. A set is called countable if it has the same size as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. Georg Cantor, the founder of set theory, proved that the set of integers and the set of rational numbers are both countable, by constructing such maps. More famously, Cantor proved that the real numbers and that the powerset of the natural numbers are both uncountable, by assuming there was a map \( f \) and deriving a contradiction. <nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using ordinal arithmetic. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki> Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the continuum hypothesis. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven. 2bc53a647462ff99fcdc70ef641f9fef01a71b57 0 68 178 2023-08-28T15:21:45Z RhubarbJayde 25 Redirected page to [[Countability]] wikitext text/x-wiki #REDIRECT [[Countability]] 83755c693a20bad68152328fe33a6068aa6e2a3f 0 9 179 56 2023-08-28T15:23:32Z RhubarbJayde 25 wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. 2bd263ad339d2a3b1d50ad4047a0154917180ede 0 11 180 36 2023-08-28T15:38:14Z RhubarbJayde 25 wikitext text/x-wiki '''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). Since the function is continuous in the order topology, they are the same as the closure points. Using the [[Veblen hierarchy]], they are enumerated as \(\varphi(1,\alpha)\). The least epsilon number is the limit of "predicative" [[Cantor normal form]], since, as we mentioned, it can't be reached from below via base-\(\omega\) exponentiation. And, in general, \(\varphi(1,\alpha+1)\) is the least ordinal that can't be reached from \(\varphi(1,\alpha)\). By Veblen's fixed point lemma, the enumerating function of the epsilon numbers is normal and thus also has fixed points - these are denoted \(\varphi(2,\alpha)\) or \(\zeta_\alpha\). By iterating Cantor normal form and the process of taking (common) fixed points, the [[Veblen hierarchy]] is formed. This induces a natural normal form, called Veblen normal form. Its limit is not \(\zeta_0\), but a much larger ordinal, denoted \(\Gamma_0\). And in general, the ordinals that can't be obtained from below via Veblen normal form are called strongly critical. They are important in ordinal analysis. __NOEDITSECTION__ <!-- Remove the section edit links --> c39dd43bdebe658f293e86055dc8ccfb5ec25b45 0 56 181 142 2023-08-28T15:55:22Z RhubarbJayde 25 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \). Ordinals beyond \( \Gamma_0 \) can either be written using a variadic extension of the Veblen hierarchy, or using ordinal collapsing functions. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). 996bd7efa7fc22c1487caf1e16a03a044089f6b1 0 69 182 2023-08-28T17:42:04Z RhubarbJayde 25 Created page with "The patterns of resemblance (PoR) are a system of ordinal-notations introduced by TJ Carlson. It is superficially similar to stability, yet is a notation for recursive rather than nonrecursive ordinals, and uses elementary substructures between ordinals themselves, instead of between ranks of the constructible universe. It uses a structure also found in [[Bashicu matrix system|BMS]] known as respecting forests, and was originally believed to have the same limit as BMS." wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by TJ Carlson. It is superficially similar to stability, yet is a notation for recursive rather than nonrecursive ordinals, and uses elementary substructures between ordinals themselves, instead of between ranks of the constructible universe. It uses a structure also found in [[Bashicu matrix system|BMS]] known as respecting forests, and was originally believed to have the same limit as BMS. ac0172e871218c748c406c12e6f5fdee193fddbe 201 182 2023-08-29T21:56:32Z C7X 9 Stability isn't an ordinal notation wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by TJ Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. af75169f9097c7128f45388aa324624df8968330 202 201 2023-08-29T22:18:24Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref>T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref>https://arxiv.org/pdf/1710.01870.pdf</ref> For all systems analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "Elementary patterns of resemblance", corollary 6.12. Annals of Pure and Applied Logic vol. 108 (2001), pp.19--77.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> 9598a2d10ddb5725b8629626037a343051dc82bb 203 202 2023-08-29T22:31:15Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref>T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997), implicit in section 3. Accessed 29 August 2023.</ref><ref>T. J. Carlson, "Elementary patterns of resemblance", corollary 6.12. Annals of Pure and Applied Logic vol. 108 (2001), pp.19--77.</ref><ref>G. Wilken, "[ Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref>G. Wilken, [https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2] (2017). Accessed 29 August 2023.</ref>^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) 2eb3cb182eedf7d5cef102b111bb165966847edd 204 203 2023-08-29T22:38:40Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997), implicit in section 3. Accessed 29 August 2023.</ref><ref>T. J. Carlson, "Elementary patterns of resemblance", corollary 6.12. Annals of Pure and Applied Logic vol. 108 (2001), pp.19--77.</ref><ref>G. Wilken, "[ Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref>G. Wilken, [https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2] (2017). Accessed 29 August 2023.</ref>^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}_L\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. 9a0656438bcb7313c043a6dd8e2df9f5314c8ca7 205 204 2023-08-29T22:38:52Z C7X 9 /* Stability */ wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997), implicit in section 3. Accessed 29 August 2023.</ref><ref>T. J. Carlson, "Elementary patterns of resemblance", corollary 6.12. Annals of Pure and Applied Logic vol. 108 (2001), pp.19--77.</ref><ref>G. Wilken, "[ Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref>G. Wilken, [https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2] (2017). Accessed 29 August 2023.</ref>^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. 0b75a1f2f0b684488a1a36c48624c54bc5c672f2 0 70 183 2023-08-29T17:29:21Z 82.8.204.174 0 Created page with "The infinite time Turing machines are a powerful method of computation introduced by Joel David Hamkins and Andy Lewis.<ref>Infinite Time Turing Machines, Joel David Hamkins and Andy Lewis, 1998</ref> They augment the normal notion of a Turing machine (first introduced by Alan Turing in his seminal paper<ref>Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". ''Proceedings of the London Mathematical Society''.</ref>), to a hypot..." wikitext text/x-wiki The infinite time Turing machines are a powerful method of computation introduced by Joel David Hamkins and Andy Lewis.<ref>Infinite Time Turing Machines, Joel David Hamkins and Andy Lewis, 1998</ref> They augment the normal notion of a Turing machine (first introduced by Alan Turing in his seminal paper<ref>Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". ''Proceedings of the London Mathematical Society''.</ref>), to a hypothetical machine model which can run for infinitely many steps. It separates the tape into three separate tapes - the input, scratch and output tapes. It's possible to define an analogue of the busy beaver function for ITTMs, denoted \(\Sigma_\infty\), which grows significantly faster than the ordinary busy beaver function, even with an oracle, as well as a halting problem for ITTMs, which has higher Turing degree than \(0^{(\alpha)}\) for all \(\alpha < \gamma\), where \(\gamma\) is the second of the following large ITTM-related ordinals: * \(\lambda\) is the supremum of all ordinals which are the output of an ITTM with empty input. * \(\gamma\) is the supremum of all halting times of an ITTM with empty input. * \(\zeta\) is the supremum of all ordinals which are the eventual output of an ITTM with empty input. * \(\Sigma\) is the supremum of all ordinals which are the accidental output of an ITTM with empty input. The ITTM theorem says that \(\lambda = \gamma\), and that: * \(\lambda\) is \(\zeta\)-stable (i.e. \(\zeta\)-\(\Sigma_1\)-stable). * \(\zeta\) is the least ordinal which is \(\rho\)-\(\Sigma_2\)-stable for some \(\rho > \zeta\). * \(\Sigma\) is the least ordinal so that \(\zeta\) is \(\Sigma\)-\(\Sigma_2\)-stable. This further shows the computational potency of ITTMs, since the limit of the order-types of well-orders they can compute is much greater than that of normal TMs, i.e. \(\omega_1^{\mathrm{CK}\). 7487c30ea9190a582cd5d220bc922090b2a97b60 184 183 2023-08-29T17:29:52Z 82.8.204.174 0 wikitext text/x-wiki The infinite time Turing machines are a powerful method of computation introduced by Joel David Hamkins and Andy Lewis.<ref>Infinite Time Turing Machines, Joel David Hamkins and Andy Lewis, 1998</ref> They augment the normal notion of a Turing machine (first introduced by Alan Turing in his seminal paper<ref>Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". ''Proceedings of the London Mathematical Society''.</ref>), to a hypothetical machine model which can run for infinitely many steps. It separates the tape into three separate tapes - the input, scratch and output tapes. It's possible to define an analogue of the busy beaver function for ITTMs, denoted \(\Sigma_\infty\), which grows significantly faster than the ordinary busy beaver function, even with an oracle, as well as a halting problem for ITTMs, which has higher Turing degree than \(0^{(\alpha)}\) for all \(\alpha < \gamma\), where \(\gamma\) is the second of the following large ITTM-related ordinals: * \(\lambda\) is the supremum of all ordinals which are the output of an ITTM with empty input. * \(\gamma\) is the supremum of all halting times of an ITTM with empty input. * \(\zeta\) is the supremum of all ordinals which are the eventual output of an ITTM with empty input. * \(\Sigma\) is the supremum of all ordinals which are the accidental output of an ITTM with empty input. The ITTM theorem says that \(\lambda = \gamma\), and that: * \(\lambda\) is \(\zeta\)-stable (i.e. \(\zeta\)-\(\Sigma_1\)-stable). * \(\zeta\) is the least ordinal which is \(\rho\)-\(\Sigma_2\)-stable for some \(\rho > \zeta\). * \(\Sigma\) is the least ordinal so that \(\zeta\) is \(\Sigma\)-\(\Sigma_2\)-stable. <nowiki>This further shows the computational potency of ITTMs, since the limit of the order-types of well-orders they can compute is much greater than that of normal TMs, i.e. \(\omega_1^{\mathrm{CK}}\).</nowiki> a614c40ebab6c8fd99ead24fa0a8d1a2e9f88291 185 184 2023-08-29T17:55:04Z RhubarbJayde 25 wikitext text/x-wiki The infinite time Turing machines are a powerful method of computation introduced by Joel David Hamkins and Andy Lewis.<ref>Infinite Time Turing Machines, Joel David Hamkins and Andy Lewis, 1998</ref> They augment the normal notion of a Turing machine (first introduced by Alan Turing in his seminal paper<ref>Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". ''Proceedings of the London Mathematical Society''.</ref>), to a hypothetical machine model which can run for infinitely many steps. It separates the tape into three separate tapes - the input, scratch and output tapes. It's possible to define an analogue of the busy beaver function for ITTMs, denoted \(\Sigma_\infty\), which grows significantly faster than the ordinary busy beaver function, even with an oracle, as well as a halting problem for ITTMs, which has higher Turing degree than \(0^{(\alpha)}\) for all \(\alpha < \gamma\), where \(\gamma\) is the second of the following large ITTM-related ordinals: * \(\lambda\) is the supremum of all ordinals which are the output of an ITTM with empty input. * \(\gamma\) is the supremum of all halting times of an ITTM with empty input. * \(\zeta\) is the supremum of all ordinals which are the eventual output of an ITTM with empty input. * \(\Sigma\) is the supremum of all ordinals which are the accidental output of an ITTM with empty input. The ITTM theorem says that \(\lambda = \gamma\), and that: * \(\lambda\) is \(\zeta\)-stable (i.e. \(\zeta\)-\(\Sigma_1\)-stable). * \(\zeta\) is the least ordinal which is \(\rho\)-\(\Sigma_2\)-stable for some \(\rho > \zeta\). * \(\Sigma\) is the least ordinal so that \(\zeta\) is \(\Sigma\)-\(\Sigma_2\)-stable. <nowiki>This further shows the computational potency of ITTMs, since the limit of the order-types of well-orders they can compute is much greater than that of normal TMs, i.e. \(\omega_1^{\mathrm{CK}}\).</nowiki> Infinite time Turing machines can themselves be generalized further to \(\Sigma_n\)-machines, with \(\Sigma_2\)-machines being the same as the original.<ref>Friedman, Sy-David & Welch, P. D. (2011). Hypermachines. Journal of Symbolic Logic</ref> a1991c2bf84369af03998b730e7dbab3cf4ba039 0 71 188 2023-08-29T18:22:55Z RhubarbJayde 25 Created page with "A weakly compact cardinal is a certain kind of [[large cardinal]]. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indes..." wikitext text/x-wiki A weakly compact cardinal is a certain kind of [[large cardinal]]. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indescribable. * \(\kappa\) is \(\kappa\)-unfoldable. * The partition property \(\kappa \to (\kappa)^2_2\) holds. Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem. The existence of a weakly compact cardinal is not provable in ZFC - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, the least weakly compact is still smaller than a lot of other large cardinals, such as totally reflecting cardinals. 52a6e76ed881413807519a829b3878c01e37793d 194 188 2023-08-29T21:24:28Z C7X 9 wikitext text/x-wiki A weakly compact cardinal is a certain kind of [[large cardinal]]. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indescribable. * \(\kappa\) is \(\kappa\)-unfoldable. * The partition property \(\kappa \to (\kappa)^2_2\) holds. Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem. The existence of a weakly compact cardinal is not provable in ZFC - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, since "\(\kappa\) is weakly compact" is a \(\Pi^1_2\) property of \(V_\kappa\),<ref>Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:[https://doi.org/10.1007%2F978-3-540-88867-3_2 10.1007/978-3-540-88867-3_2]. ISBN 3-540-00384-3</ref> i.e. a \(\Pi_2\) property of \(V_{\kappa+1}\),<ref>J. D. Hamkins, "[https://jdh.hamkins.org/local-properties-in-set-theory/ Local properties in set theory]" (2014), blog post. Accessed 29 August 2023.</ref> a totally reflecting cardinal is larger than the least weakly compact cardinal. ==References== <references /> 815edffefb842c0975e1a8bfb60cb361cf43663f 0 51 189 128 2023-08-29T19:26:54Z RhubarbJayde 25 wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order-type to any well-ordered set. The order-type of a well-ordered set is intuitively its "length". In particular, the order-type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order-type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order-type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) There are helpful visual representations for these, namely with sets of lines. For some basic intuition, [https://www.youtube.com/watch?v=SrU9YDoXE88 see this video]. For example, \(\alpha + \beta\) can be visualized as \(\alpha\), followed by a copy of \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This is formally proved by the following: "if \(X\) and \(Y\) are well-ordered sets with order-types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order-type \(\alpha + \beta\)". Also, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order-type: the order-type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. 8f9aa676d28dc8fba4afe48322be6a7a02f18800 199 189 2023-08-29T21:50:34Z C7X 9 Provable over ZFC (and I'd expect some weaker theories) wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of well-ordered sets. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order-type to any well-ordered set. The order-type of a well-ordered set is intuitively its "length". In particular, the order-type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order-type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order-type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) There are helpful visual representations for these, namely with sets of lines. For some basic intuition, [https://www.youtube.com/watch?v=SrU9YDoXE88 see this video]. For example, \(\alpha + \beta\) can be visualized as \(\alpha\), followed by a copy of \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over ZFC: "if \(X\) and \(Y\) are well-ordered sets with order-types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order-type \(\alpha + \beta\)". Also, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order-type: the order-type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. d823998945a0e9be690877ed2674ad990da90e35 1 72 190 2023-08-29T21:15:02Z C7X 9 /* "A relatively convoluted definition" */ new section wikitext text/x-wiki == "A relatively convoluted definition" == In my opinion as I've been working more with infinitary languages, it seems like the compactness property of weakly compact cardinals is not very complicated. The compactness theorem already admits some extensions, with the most attention directed to what to replace the word "finite" in the compactness theorem with. Then there is only one major change between the compactness theorem and the weak compactness property: * When Γ is a set of L_ω,ω-sentences ''of size &lt;&alefsym;₀'', if every subset of Γ of size &lt;&alefsym;₀ has a model, then Γ has a model. * When Γ is a set of L_κ,κ-sentences of size &lt;κ, if every subset of Γ of size &lt;κ has a model, then Γ has a model. where the change is the italicized part. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 21:15, 29 August 2023 (UTC) a9f79341989ac22658a1352c6f180fffd030d597 191 190 2023-08-29T21:17:17Z C7X 9 wikitext text/x-wiki == "A relatively convoluted definition" == In my opinion as I've been working more with infinitary languages, it seems like the compactness property of weakly compact cardinals is not very complicated. The compactness theorem already admits some extensions, with the most attention directed to what to replace the word "finite" in the compactness theorem with. Then there is only one major change between the compactness theorem and the weak compactness property: * When \(\Gamma\) is a set of \(\mathcal L_{\omega,\omega}\)-sentences <u>of size \(<\aleph_0\)</u>, if every subset of \(\Gamma\) of size \(<\aleph_0\) has a model, then \(\Gamma\) has a model. * When \(\Gamma\) is a set of \(\mathcal L_{\kappa,\kappa}\)-sentences of size \(<\kappa\), if every subset of \(\Gamma\) of size \(<\kappa\) has a model, then \(\Gamma\) has a model. where the change is the italicized part. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 21:15, 29 August 2023 (UTC) 7cd45eb1fbd35ce27013bb4d461561ae260e33ad 192 191 2023-08-29T21:17:33Z C7X 9 /* "A relatively convoluted definition" */ wikitext text/x-wiki == "A relatively convoluted definition" == In my opinion as I've been working more with infinitary languages, it seems like the compactness property of weakly compact cardinals is not very complicated. The compactness theorem already admits some extensions, with the most attention directed to what to replace the word "finite" in the compactness theorem with. Then there is only one major change between the compactness theorem and the weak compactness property: * When \(\Gamma\) is a set of \(\mathcal L_{\omega,\omega}\)-sentences <u>of size \(\underline{<\aleph_0}\)</u>, if every subset of \(\Gamma\) of size \(<\aleph_0\) has a model, then \(\Gamma\) has a model. * When \(\Gamma\) is a set of \(\mathcal L_{\kappa,\kappa}\)-sentences of size \(<\kappa\), if every subset of \(\Gamma\) of size \(<\kappa\) has a model, then \(\Gamma\) has a model. where the change is the underlined part. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 21:15, 29 August 2023 (UTC) feb4e86597a85759fb5c5c2ada78fe20a0cf095d 193 192 2023-08-29T21:18:06Z C7X 9 /* "A relatively convoluted definition" */ wikitext text/x-wiki == "A relatively convoluted definition" == In my opinion as I've been working more with infinitary languages, it seems like the compactness property of weakly compact cardinals is not very complicated. The compactness theorem already admits some extensions, with the most attention directed to what to replace the word "finite" in the compactness theorem with. Then there is only one major change between the compactness theorem and the weak compactness property: * When \(\Gamma\) is a set of \(\mathcal L_{\omega,\omega}\)-sentences <u>of size \(\underline{<\aleph_0}\)</u>, if every subset of \(\Gamma\) of size \(<\aleph_0\) has a model, then \(\Gamma\) has a model. * When \(\Gamma\) is a set of \(\mathcal L_{\kappa,\kappa}\)-sentences of size \(<\kappa\), if every subset of \(\Gamma\) of size \(<\kappa\) has a model, then \(\Gamma\) has a model. where the change is the underlined part. Unless there is a source for it I am not sure about this statement. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 21:15, 29 August 2023 (UTC) eaebc6aafc9b61ddddb25e3d0297f952e641c4f7 0 53 195 130 2023-08-29T21:26:00Z C7X 9 wikitext text/x-wiki '''Fodor's lemma''' (or the '''pressing-down lemma''') is a lemma proven by Géza Fodor in 1956. The lemma states that when \(\kappa\) is an uncountable regular cardinal and \(S\) is a stationary set of ordinals \(<\kappa\), any regressive function \(f:S\to\{<\kappa\}\) must be constant on a stationary set of ordinals \(<\kappa\). ==Importance to apierology== Since \(\omega_1\) is regular, setting \(S=\{<\omega_1\}\), Fodor's lemma implies there does not exist a fundamental sequence system that both assigns a sequence to all countable ordinals, and has the Bachmann property. Fodor's lemma may fail without choice<ref>A. Karagila, [https://arxiv.org/abs/1610.03985 Fodor's Lemma can Fail Everywhere]</ref>, but instead we can use a weakened version known as '''Neumer's theorem''' to prove this result, in which the set \(S\) is removed and replaced with \(\{<\kappa\}\) in all cases. ==References== * E. Tachtsis, [https://www.ams.org/journals/proc/2020-148-03/S0002-9939-2019-14794-8/S0002-9939-2019-14794-8.pdf Juh&aacute;sz's topological generalization of Neumer's theorem may fail in ZF] (2019). Corollary 2.7. * Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3. <references /> af5a462a1718b69ccfc6350fb221e42b371c6619 0 60 196 162 2023-08-29T21:29:25Z C7X 9 Possibly clearer wording wikitext text/x-wiki A '''sequence system''' is an [[ordinal notation system]] in which the terms of the notation are sequences. Typically, it is an [[expansion system]], with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> Notable sequence systems include [[Primitive sequence system]], [[Pair sequence system]], [[Sudden sequence system]], [[Bashicu matrix system]] and [[Y sequence]]. 3c9a66819a585fb4b24ad9a8ea6522b99f7edd1c 0 73 198 2023-08-29T21:40:36Z C7X 9 Created page with "A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\se..." wikitext text/x-wiki A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).<ref name="MarekSrebrny73" />^p.368 ==Longer gaps== Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73" />^p.365 4404d44a7093ec4f9c8bc1117a9f68d88b3b7418 0 48 200 133 2023-08-29T21:51:45Z C7X 9 Linking wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]]<ref>D. Probst, [https://boris.unibe.ch/108693/1/pro17.pdf#page=153 A modular ordinal analysis of metapredicative subsystems of second-order arithmetic] (2017), p.153</ref> or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 48675dad9ddcd27ed39030602b9c3127c061c2e1 0 74 206 2023-08-30T09:13:52Z 24.43.123.71 0 Created page with "The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes [[zero]]. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite [[ordinal]]s. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of e..." wikitext text/x-wiki The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes [[zero]]. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite [[ordinal]]s. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. ===Von Neumann ordinals=== In [[ZFC]] and other set theories without urelements, we can define natural numbers by applying the definition of Von Neumann ordinals to finite ordinals. In this view, each natural number is the set of previous naturals: \( 0 = \varnothing \) and \( n + 1 = \{0, \dots, n\} \). ===Zermelo ordinals=== [[:wikipedia:Ernst Zermelo|Ernst Zermelo]] provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). ===Frege and Russell=== [[:wikipedia:Gottlob Frege|Gottlob Frege]] and [[:wikipedia:Bertrand Russell|Bertrand Russell]] proposed defining a natural number \( n \) as the equivalence [[class]] of all sets with [[cardinality]] \( n \). This definition cannot be realized in ZFC, because the classes involved are [[proper class]]es. ===Church numerals=== In the [[lambda calculus]], the standard way to encode natural numbers is as Church numerals, developed by [[:wikipedia:Alonzo Church|Alonzo Church]]. In this encoding, each natural number \( n \) is identified with a function that returns the composition of its input with itself \( n \) times: \( 0 := \lambda f. \lambda x. x \), \( 1 := \lambda f. \lambda x. f x \), \( 2 := \lambda f. \lambda x. f (f x) \), etc. ==Theories of arithmetic== Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]]. 8c40637aa522feebb61fa47efc6a6424559fa30f 207 206 2023-08-30T09:14:38Z 24.43.123.71 0 wikitext text/x-wiki The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes [[zero]]. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite [[ordinal]]s. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. ===Von Neumann ordinals=== In [[ZFC]] and other set theories without urelements, we can define natural numbers by applying the definition of Von Neumann ordinals to finite ordinals. In this view, each natural number is the set of previous naturals: \( 0 = \varnothing \) and \( n + 1 = \{0, \dots, n\} \). ===Zermelo ordinals=== [[:wikipedia:Ernst Zermelo|Ernst Zermelo]] provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). Unlike the Von Neumann ordinals, Zermelo's encoding can only be used to represent finite ordinals. ===Frege and Russell=== [[:wikipedia:Gottlob Frege|Gottlob Frege]] and [[:wikipedia:Bertrand Russell|Bertrand Russell]] proposed defining a natural number \( n \) as the equivalence [[class]] of all sets with [[cardinality]] \( n \). This definition cannot be realized in ZFC, because the classes involved are [[proper class]]es. ===Church numerals=== In the [[lambda calculus]], the standard way to encode natural numbers is as Church numerals, developed by [[:wikipedia:Alonzo Church|Alonzo Church]]. In this encoding, each natural number \( n \) is identified with a function that returns the composition of its input with itself \( n \) times: \( 0 := \lambda f. \lambda x. x \), \( 1 := \lambda f. \lambda x. f x \), \( 2 := \lambda f. \lambda x. f (f x) \), etc. ==Theories of arithmetic== Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]]. 8b22c98c40e8d2dcefbf0eb2c5e54cddbd05c287 209 207 2023-08-30T10:27:50Z RhubarbJayde 25 wikitext text/x-wiki The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes [[zero]]. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite [[ordinal]]s. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. In many areas such as geometry or number theory, they are taken as primitives and not formally treated ===Von Neumann ordinals=== In [[ZFC]] and other set theories without urelements, we can define natural numbers by applying the definition of Von Neumann ordinals to finite ordinals. In this view, each natural number is the set of previous naturals: \( 0 = \varnothing \) and \( n + 1 = \{0, \dots, n\} \). ===Zermelo ordinals=== [[:wikipedia:Ernst Zermelo|Ernst Zermelo]] provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). Unlike the Von Neumann ordinals, Zermelo's encoding can only be used to represent finite ordinals. ===Frege and Russell=== During the early development of foundational philosophy and logicism, [[:wikipedia:Gottlob Frege|Gottlob Frege]] and [[:wikipedia:Bertrand Russell|Bertrand Russell]] proposed defining a natural number \( n \) as the equivalence [[class]] of all sets with [[cardinality]] \( n \). This definition cannot be realized in ZFC, because the classes involved are [[proper class]]es, except for \( n = 0 \). ===Church numerals=== In the [[lambda calculus]], the standard way to encode natural numbers is as Church numerals, developed by [[:wikipedia:Alonzo Church|Alonzo Church]]. In this encoding, each natural number \( n \) is identified with a function that returns the composition of its input with itself \( n \) times: \( 0 := \lambda f. \lambda x. x \), \( 1 := \lambda f. \lambda x. f x \), \( 2 := \lambda f. \lambda x. f (f x) \), etc. ==Theories of arithmetic== Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]]. 10cbd4b4e22164b475f52d67d6b0ebfc05b92dea 0 75 208 2023-08-30T09:14:55Z 24.43.123.71 0 Redirected page to [[0]] wikitext text/x-wiki #REDIRECT [[0]] e700d17623dbc9b3e6b83a0ab9ba7fff96b62f01 0 76 210 2023-08-30T10:28:42Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 77 211 2023-08-30T10:29:04Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 78 212 2023-08-30T10:29:46Z RhubarbJayde 25 Redirected page to [[Epsilon numbers]] wikitext text/x-wiki #REDIRECT [[Epsilon numbers]] 4603d418f29680a0e3b925e67b4471f50e268bb4 0 79 213 2023-08-30T10:30:07Z RhubarbJayde 25 Redirected page to [[Epsilon numbers]] wikitext text/x-wiki #REDIRECT [[Epsilon numbers]] 4603d418f29680a0e3b925e67b4471f50e268bb4 0 80 214 2023-08-30T10:30:28Z RhubarbJayde 25 Redirected page to [[Epsilon numbers]] wikitext text/x-wiki #REDIRECT [[Epsilon numbers]] 4603d418f29680a0e3b925e67b4471f50e268bb4 0 81 215 2023-08-30T10:30:59Z RhubarbJayde 25 Redirected page to [[Epsilon numbers]] wikitext text/x-wiki #REDIRECT [[Epsilon numbers]] 4603d418f29680a0e3b925e67b4471f50e268bb4 0 82 216 2023-08-30T10:31:22Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 56 217 181 2023-08-30T10:33:51Z RhubarbJayde 25 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \). Ordinals beyond \( \Gamma_0 \) can either be written using a variadic extension of the Veblen hierarchy, or using ordinal collapsing functions. Ordinals unreachable from below via Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). a329b8eab30c0188fbf680e52a6ccc2afc6f26b5 230 217 2023-08-30T11:14:50Z RhubarbJayde 25 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals beyond \( \Gamma_0 \) can either be written using a variadic extension of the Veblen hierarchy (which can also be used to define the [[Small Veblen ordinal|SVO]] and [[Large Veblen ordinal|LVO]]), or using ordinal collapsing functions. Ordinals unreachable from below via Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). 9716cf37783d6d12a1eaedbea9cc183456349eb6 0 83 218 2023-08-30T10:39:58Z RhubarbJayde 25 Created page with "The Small Veblen ordinal is the limit of a finitary, variadic extension of the [[Veblen hierarchy]]. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. [[Veblen hierarchy|strongly critical ordinals]] - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least..." wikitext text/x-wiki The Small Veblen ordinal is the limit of a finitary, variadic extension of the [[Veblen hierarchy]]. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. [[Veblen hierarchy|strongly critical ordinals]] - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least ordinal not reachable from below via this function, namely the limit of \( \omega \), \( \varepsilon_0 \), \( \Gamma_0 \), \( \varphi(1,0,0,0) \) (the Ackermann ordinal), ... In ordinal collapsing functions, in particular Buchholz's psi function, it is considered the countable collapse of \( \Omega^{\Omega^\omega} \), and may be denoted by \( \psi_0(\Omega^{\Omega^\omega}) \). dda9cccfb2ba37bc3034ca406a9644badda200b3 0 84 219 2023-08-30T10:40:35Z RhubarbJayde 25 Redirected page to [[Small Veblen ordinal]] wikitext text/x-wiki #REDIRECT [[Small Veblen ordinal]] f97cb32c0be6c257f9958b490d9200ad9f5a6a39 0 85 220 2023-08-30T10:44:40Z RhubarbJayde 25 Created page with "The large Veblen ordinal is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,...,0,0) \), with \( \omega \) many zeroes. The..." wikitext text/x-wiki The large Veblen ordinal is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,...,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi(1 @ \alpha) \), where \( 1 @ \alpha \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). 9cf3094ca328dd53318e3f0eeadf251b0686e2eb 221 220 2023-08-30T10:46:02Z RhubarbJayde 25 wikitext text/x-wiki The large Veblen ordinal is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,...,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi(1 @ \alpha) \), where \( 1 @ \alpha \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi(1 @ (1,0)) \), which represents the large Veblen ordinal. This system's limit is the [[Bachmann-Howard ordinal]]. a9c40e6f947df3f4e11e5e19d1492586fc626519 0 86 222 2023-08-30T10:51:28Z RhubarbJayde 25 Created page with "The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek s..." wikitext text/x-wiki The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). fcad961bbc12a27be3077f14886a61a7da966e95 224 222 2023-08-30T10:52:38Z RhubarbJayde 25 wikitext text/x-wiki The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). Buchholz further extended this to his famous set of collapsing functions, whose limit is the much larger [[Buchholz ordinal]]. c51356bc5a5ef5f219580297794ed2d5640e5efb 0 87 223 2023-08-30T10:51:50Z RhubarbJayde 25 Redirected page to [[Large Veblen ordinal]] wikitext text/x-wiki #REDIRECT [[Large Veblen ordinal]] d0dc149bdfe38ef71c4d89c9c2ecdf8b6aabe547 0 88 225 2023-08-30T11:03:55Z RhubarbJayde 25 Created page with "The Buchholz ordinal is the limit of Wilfried Buchholz's original set of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), [[Bachmann-Howard ordinal|\( \mathrm{BHO} \)]], \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, or of Peano arith..." wikitext text/x-wiki The Buchholz ordinal is the limit of Wilfried Buchholz's original set of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), [[Bachmann-Howard ordinal|\( \mathrm{BHO} \)]], \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, or of Peano arithmetic with finitely iterated inductive definitions, as well as the limit of pair sequence system in [[Bashicu matrix system]]. It can be viewed as an extension of the Bachmann-Howard ordinal by allowing higher uncountable cardinals, which collapse above \( \Omega \). 21e0aaa3be415249b08ee44e9cabe2eb3ae71b8a 226 225 2023-08-30T11:04:14Z RhubarbJayde 25 wikitext text/x-wiki The Buchholz ordinal is the limit of Wilfried Buchholz's original set of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), [[Bachmann-Howard ordinal|\( \mathrm{BHO} \)]], \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, or of Peano arithmetic with finitely iterated inductive definitions, as well as the limit of pair sequence system in [[Bashicu matrix system]]. It can be viewed as an extension of the Bachmann-Howard ordinal by allowing higher uncountable cardinals, which iteratively collapse above \( \Omega \). 9d931c99a87221684ec3df650ad0bd2a39fe938e 0 89 227 2023-08-30T11:11:03Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 90 228 2023-08-30T11:11:27Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 91 229 2023-08-30T11:11:56Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 92 231 2023-08-30T11:15:28Z RhubarbJayde 25 Redirected page to [[Small Veblen ordinal]] wikitext text/x-wiki #REDIRECT [[Small Veblen ordinal]] f97cb32c0be6c257f9958b490d9200ad9f5a6a39 0 93 232 2023-08-30T11:15:54Z RhubarbJayde 25 Redirected page to [[Large Veblen ordinal]] wikitext text/x-wiki #REDIRECT [[Large Veblen ordinal]] d0dc149bdfe38ef71c4d89c9c2ecdf8b6aabe547 0 94 233 2023-08-30T11:16:30Z RhubarbJayde 25 Redirected page to [[Bachmann-Howard ordinal]] wikitext text/x-wiki #REDIRECT [[Bachmann-Howard ordinal]] 03d2a8c79515ff9f4907a83c444918244bbfbbab 0 95 234 2023-08-30T11:17:06Z RhubarbJayde 25 Redirected page to [[Extended Buchholz ordinal]] wikitext text/x-wiki #REDIRECT [[Extended Buchholz ordinal]] 38daf2b7770537c4bfd2009af817c56b4b976a1a 0 96 235 2023-08-30T11:27:02Z RhubarbJayde 25 Created page with "The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the ..." wikitext text/x-wiki The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \( \Pi^1_1 \)-formulae (of which the Buchholz ordinal is the proof-theoretic ordinal) with an additional scheme of transfinite induction. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). 10484ae284928ff7e8598e0b8946749d5b3f9cee 0 97 236 2023-08-30T11:28:40Z RhubarbJayde 25 Redirected page to [[Bird ordinal]] wikitext text/x-wiki #REDIRECT [[Bird ordinal]] 0d1d12370eaa7af33f8dcd00547e4bcd10cf6a39 0 98 237 2023-08-30T11:31:16Z RhubarbJayde 25 Created page with "The Extended Buchholz ordinal, sometimes known as OFP (short for omega-fixed-point), is the limit of an extension of Buchholz's original set of ordinal collapsing functions, defined by Denis Maksudov, which allows to collapse ordinals such as \( \Omega_{\omega + 1} \) (which corresponds to the [[Takeuti-Feferman-Buchholz ordinal]]), \( \Omega_{\omega^2} \) (which is believed to correspond to the [[Bashicu matrix system|BMS]] matrix (0,0,0)(1,1,1)(2,1,1)), or \( \Omega_{\..." wikitext text/x-wiki The Extended Buchholz ordinal, sometimes known as OFP (short for omega-fixed-point), is the limit of an extension of Buchholz's original set of ordinal collapsing functions, defined by Denis Maksudov, which allows to collapse ordinals such as \( \Omega_{\omega + 1} \) (which corresponds to the [[Takeuti-Feferman-Buchholz ordinal]]), \( \Omega_{\omega^2} \) (which is believed to correspond to the [[Bashicu matrix system|BMS]] matrix (0,0,0)(1,1,1)(2,1,1)), or \( \Omega_{\Omega} \) (which corresponds to the [[Bird ordinal]]). It has not been widely studied in the literature, but is common in amateur apeirological discussions, and is known to correspond to the proof-theoretic ordinal of \( \PI^1_1 \mathrm{-TR}_0 \), a strengthening of arithmetical transfinite recursion (the proof theoretic ordinal of which is the [[Feferman-Schütte ordinal]]). Thus, one could claim that it is to the [[Buchholz ordinal]] as the [[Feferman-Schütte ordinal]] is to [[Epsilon numbers|\( \varepsilon_0 \)]]. 41dc2fa51d9cea1e7357d093a4ebc5a33add7fc0 238 237 2023-08-30T11:31:33Z RhubarbJayde 25 wikitext text/x-wiki The Extended Buchholz ordinal, sometimes known as OFP (short for omega-fixed-point), is the limit of an extension of Buchholz's original set of ordinal collapsing functions, defined by Denis Maksudov, which allows to collapse ordinals such as \( \Omega_{\omega + 1} \) (which corresponds to the [[Takeuti-Feferman-Buchholz ordinal]]), \( \Omega_{\omega^2} \) (which is believed to correspond to the [[Bashicu matrix system|BMS]] matrix (0,0,0)(1,1,1)(2,1,1)), or \( \Omega_{\Omega} \) (which corresponds to the [[Bird ordinal]]). It has not been widely studied in the literature, but is common in amateur apeirological discussions, and is known to correspond to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-TR}_0 \), a strengthening of arithmetical transfinite recursion (the proof theoretic ordinal of which is the [[Feferman-Schütte ordinal]]). Thus, one could claim that it is to the [[Buchholz ordinal]] as the [[Feferman-Schütte ordinal]] is to [[Epsilon numbers|\( \varepsilon_0 \)]]. f097e5913df003eeadb3bb3fcbeb1d5d6a13ec7c 0 99 239 2023-08-30T11:31:51Z RhubarbJayde 25 Redirected page to [[Bashicu matrix system]] wikitext text/x-wiki #REDIRECT [[Bashicu matrix system]] f53be39a19b5726f797ceacf0383343641b74849 0 100 240 2023-08-30T11:43:28Z RhubarbJayde 25 Created page with "The Bird ordinal (sometimes called Bird's ordinal) is an intermediate ordinal between the [[Takeuti-Feferman-Buchholz ordinal]] and [[Extended Buchholz ordinal]] which occurs occasionally in apeirological notations such as [[Bashicu matrix system|BMS]]. It was named by the apeirological community in honor of Chris Bird. This is because it is believed to correspond to the limit of his final system of array notations, and thus the growth rate of a natural extension of his..." wikitext text/x-wiki The Bird ordinal (sometimes called Bird's ordinal) is an intermediate ordinal between the [[Takeuti-Feferman-Buchholz ordinal]] and [[Extended Buchholz ordinal]] which occurs occasionally in apeirological notations such as [[Bashicu matrix system|BMS]]. It was named by the apeirological community in honor of Chris Bird. This is because it is believed to correspond to the limit of his final system of array notations, and thus the growth rate of a natural extension of his U function. It can be written as \( \psi_0(\Omega_\Omega) \) (not to be confused with [[Buchholz ordinal|\( \psi_0(\Omega_\omega) \)]]) in Denis Maksudov's extension of Wilfried Buchholz's system of ordinal collapsing functions. It is believed to correspond to the proof-theoretic ordinal of \( \mathrm{Aut}(\mathrm{ID}) \), the minimal extension of Peano arithmetic so that, if it proves transfinite induction along a recursive well-order of order-type \( \alpha \), then it also is able to deal with iterated inductive definitions of length \( \alpha \). This makes it essentially the maximal possible extension of the notion of iterated inductive definitions, without the addition of second-order schemata, and shows is much greater than the [[Takeuti-Feferman-Buchholz ordinal]], which is only able to deal with iterated inductive definitions of length \( \omega \). One could possibly consider the analogy that the Bird ordinal is to the Buchholz ordinal as \( \varphi(2,0,0) \) is to [[Feferman-Schütte ordinal|\( \Gamma_0 = \varphi(1,0,0) \)]], although this could potentially be seen as underestimating the size of the Bird ordinal. 521d720188fba8924db7f08918577f1a8660f4cb 0 101 241 2023-08-30T11:45:22Z RhubarbJayde 25 Redirected page to [[Church-Kleene ordinal]] wikitext text/x-wiki #REDIRECT [[Church-Kleene ordinal]] 8a7637fde14d70c14fd553c664e46fff6c007426 0 77 242 211 2023-08-30T11:54:06Z RhubarbJayde 25 RhubarbJayde moved page [[Stongly critical ordinal]] to [[Strongly critical ordinal]]: typo, I'm so stupid wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 102 243 2023-08-30T11:54:06Z RhubarbJayde 25 RhubarbJayde moved page [[Stongly critical ordinal]] to [[Strongly critical ordinal]]: typo, I'm so stupid wikitext text/x-wiki #REDIRECT [[Strongly critical ordinal]] a59d55c181dbf04c4816db5a58f188a313c6a561 0 103 244 2023-08-30T12:08:14Z RhubarbJayde 25 Created page with "<nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "recursive ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning the..." wikitext text/x-wiki <nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "recursive ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning there must be some ordinals which are still countable but they aren't recursive - i.e: they're so large that all well-orders they code are so complex that they are uncomputable. The least such is the Church-Kleene ordinal. Note that there is still a well-order on the natural numbers with order type \( \omega_1^{\mathrm{CK}} \) that is computable with an </nowiki>[[Infinite time Turing machine|''infinite time'' Turing machine]], since they are able to solve the halting problem for ordinary Turing machines and thus diagonalize over the recursive ordinals. Also, note that given computable well-orders on the natural numbers with order types \( \alpha \) and \( \beta \), it is possible to construct computable well-orders with order-types \( \alpha + \beta \), \( \alpha \cdot \beta \) and \( \alpha^{\beta} \) and much more, meaning that the Church-Kleene ordinal is not pathological and in fact a limit ordinal, [[Epsilon numbers|epsilon number]], [[Strongly critical ordinal|strongly critical]], and more. It has a variety of other convenient definitions. One of them has to do with the constructible hierarchy - \( \omega_1^{\mathrm{CK}} \) is the least admissible ordinal. In other words, it is the least limit ordinal \( \alpha > \omega \) so that, for any \( \Delta_0(L_\alpha) \)-definable function \( f: L_\alpha \to L_\alpha \), then, for all \( x \in L_\alpha \), \( f<nowiki>''</nowiki>x \in L_\alpha \). That is, the set of constructible sets with rank at most \( \omega_1^{\mathrm{CK}} \) is closed under taking preimages of an infinitary analogue of the primitive recursive functions. Note that this property still holds for \( \Sigma_1(L_\alpha) \)-functions, an infinitary analogue of Turing-computable functions, which makes sense, since the ordinals below \( \omega_1^{\mathrm{CK}} \) are a very robust class and closed under computable-esque functions. It is, in particular, equivalent to the statement: for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point (equivalent to a fixed point) of \( f \) below \( \alpha \). This also is a good explanation, since it shows it's a limit of epsilon numbers (and thus itself an epsilon number), of strongly critical numbers, etc. and why it's greater than recursive ordinals like the [[Bird ordinal]]. However, the case with \( \Sigma_2(L_\alpha) \)-functions produces a much stronger notion known as \( \Sigma_2 \)-admissibility. <nowiki>Note that, like how there is a fine hierarchy of recursive ordinals and functions on them, there is a fine hierarchy of nonrecursive ordinals, above \( \omega_1^{\mathrm{CK}} \), arguably richer.</nowiki> <nowiki>One thing to note is that many known ordinal collapsing functions are, or should be, \( \Sigma_1(L_{\omega_1^{\mathrm{CK}}}) \)-definable. Thus, the countable collapse is actually a recursive collapse, and replacing \( \Omega \) with \( \omega_1^{\mathrm{CK}} \) in an ordinal collapsing function is a possibility. While many authors do this, since it allows them to use structure more efficiently and not assume </nowiki>[[Large cardinal|large cardinal axioms]], more cumbersome proofs would be necessary, and this has led many authors such as Rathjen to instead opt for the traditional options, or use uncountable intermediates between countable nonrecursive fine structure and large cardinals, such as the reducibility hierarchy. d8b699eb7901028b1c05ff75de6450bbce25dad6 0 104 245 2023-08-30T12:38:52Z RhubarbJayde 25 Redirected page to [[Takeuti-Feferman-Buchholz ordinal]] wikitext text/x-wiki #REDIRECT [[Takeuti-Feferman-Buchholz ordinal]] c8fde5c61d7418ac5f17bda4d6191e7d77d0a339 0 105 246 2023-08-30T12:49:30Z RhubarbJayde 25 Created page with "There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \al..." wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \aleph_\omega \) are called limit cardinals because they can't be reached from below via finite iterations of the successor operation: if \( \kappa < \aleph_\omega \), then \( \kappa < \aleph_n \) for some \( n \), thus \( \kappa^{+(m)} < \aleph_{n + m} < \aleph_\omega \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence - thus they are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, cardinals such as \( \aleph_1 \), are unreachable from below via mechanisms such as transfinite recursion, and the limit of any countable sequence of countable ordinals is countable - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega\) acts as a suitable diagonalizer over ordinal arithmetic. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers. Notice that \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit fo any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it is vacuously. weakly inaccessible. Therefore, many authors add the condition of uncountabiltiy. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lamdba < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible in any means. Due to this difficulty and sufficiency of weakly inaccessible cardinals, as previously mentioned, strongly inaccessible cardinals are rarely used in apeirological circles. However, they are quite commonly used in the literature, due to results in the next section. == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. 044f4cb2dc9348cb77d369b17e68268cdfad57ac 247 246 2023-08-30T12:49:53Z RhubarbJayde 25 wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \aleph_\omega \) are called limit cardinals because they can't be reached from below via finite iterations of the successor operation: if \( \kappa < \aleph_\omega \), then \( \kappa < \aleph_n \) for some \( n \), thus \( \kappa^{+(m)} < \aleph_{n + m} < \aleph_\omega \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence - thus they are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, cardinals such as \( \aleph_1 \), are unreachable from below via mechanisms such as transfinite recursion, and the limit of any countable sequence of countable ordinals is countable - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega\) acts as a suitable diagonalizer over ordinal arithmetic. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers. Notice that \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit fo any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it is vacuously. weakly inaccessible. Therefore, many authors add the condition of uncountabiltiy. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible in any means. Due to this difficulty and sufficiency of weakly inaccessible cardinals, as previously mentioned, strongly inaccessible cardinals are rarely used in apeirological circles. However, they are quite commonly used in the literature, due to results in the next section. == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. eabb93a97e8e2b1144a6553ac616a7f49cbb0413 248 247 2023-08-30T12:50:28Z RhubarbJayde 25 wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \aleph_\omega \) are called limit cardinals because they can't be reached from below via finite iterations of the successor operation: if \( \kappa < \aleph_\omega \), then \( \kappa < \aleph_n \) for some \( n \), thus \( \kappa^{+(m)} < \aleph_{n + m} < \aleph_\omega \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence - thus they are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, cardinals such as \( \aleph_1 \), are unreachable from below via mechanisms such as transfinite recursion, and the limit of any countable sequence of countable ordinals is countable - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega\) acts as a suitable diagonalizer over ordinal arithmetic. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers. Notice that \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit fo any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it is vacuously. weakly inaccessible. Therefore, many authors add the condition of uncountabiltiy. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means. Due to this difficulty and sufficiency of weakly inaccessible cardinals, as previously mentioned, strongly inaccessible cardinals are rarely used in apeirological circles. However, they are quite commonly used in the literature, due to results in the next section. == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. 4ce65b544763def2b7a8934a41fa105fd38df75b 0 106 249 2023-08-30T13:17:41Z RhubarbJayde 25 Created page with "A Mahlo cardinal is a certain type of [[large cardinal]] used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less..." wikitext text/x-wiki A Mahlo cardinal is a certain type of [[large cardinal]] used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less popular in the literature, weakly Mahlo cardinals are more popular than strongly Mahlo cardinals in apeirological circles, but less popular in the literature. This is essentially due to the fact that weakly Mahlo cardinals are defined in terms of weakly inaccessibles, and strongly Mahlo cardinals are defined in terms of strongly inaccessibles. Essentially, we say a cardinal \( \kappa \) is weakly Mahlo if every club \( C \subseteq \kappa \) contains a regular cardinal. First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars. Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get \( (+1) \)-stable ordinals, and removing it altogether grants you a condition equivalent to regularity! Then, being Mahlo is obtained by adding the condition that such a closure point must be regular: in other words, \( \kappa \) is Mahlo iff. for any function \( f: \kappa \to \kappa \), there is a regular closure point of \( f \) below \( \kappa \). You can see that this is equivalent to Mahloness by noting that the set of closure points of any function is club, and any club is equal to the set of closure points of some function. Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy. == Strongly Mahlo == Strong Mahloness is obtained by replacing "contains a regular cardinal" with "contains a strongly inaccessible cardinal". Clearly any strongly Mahlo cardinal is weakly Mahlo, since every strongly inaccessible cardinal is regular, and the results above can be generalized to show any strongly Mahlo cardinal is strongly hyper-inaccessible. Like the situation between weakly and strongly inaccessible cardinals, \( \mathrm{GCH} \) implies weakly and strongly Mahlo cardinals are the same, while other axioms imply that \( 2^{\aleph_0} \) can be weakly Mahlo. 9d1f3e0ac41443fbc16a30a47be879fd96084080 250 249 2023-08-30T13:30:16Z RhubarbJayde 25 wikitext text/x-wiki A Mahlo cardinal is a certain type of [[large cardinal]] used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less popular in the literature, weakly Mahlo cardinals are more popular than strongly Mahlo cardinals in apeirological circles, but less popular in the literature. This is essentially due to the fact that weakly Mahlo cardinals are defined in terms of weakly inaccessibles, and strongly Mahlo cardinals are defined in terms of strongly inaccessibles. Essentially, we say a cardinal \( \kappa \) is weakly Mahlo if every club \( C \subseteq \kappa \) contains a regular cardinal. First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{\lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars. Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get \( (+1) \)-stable ordinals, and removing it altogether grants you a condition equivalent to regularity! Then, being Mahlo is obtained by adding the condition that such a closure point must be regular: in other words, \( \kappa \) is Mahlo iff. for any function \( f: \kappa \to \kappa \), there is a regular closure point of \( f \) below \( \kappa \). You can see that this is equivalent to Mahloness by noting that the set of closure points of any function is club, and any club is equal to the set of closure points of some function. Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy. == Strongly Mahlo == Strong Mahloness is obtained by replacing "contains a regular cardinal" with "contains a strongly inaccessible cardinal". Clearly any strongly Mahlo cardinal is weakly Mahlo, since every strongly inaccessible cardinal is regular, and the results above can be generalized to show any strongly Mahlo cardinal is strongly hyper-inaccessible. Like the situation between weakly and strongly inaccessible cardinals, \( \mathrm{GCH} \) implies weakly and strongly Mahlo cardinals are the same, while other axioms imply that \( 2^{\aleph_0} \) can be weakly Mahlo. == Ord is Mahlo == "Ord is Mahlo" is an assertion that, as one can likely guess, asserts that every function \( f: \mathrm{Ord} \to \mathrm{Ord} \) has a strongly inaccessible closure point. Clearly, "Ord is Mahlo" implies that there is a proper class of inaccessible cardinals, 1-inaccessible cardinals, and more. However, if \( \kappa \) is Mahlo, then \( V_\kappa \) satisfies "Ord is Mahlo", and thus "Ord is Mahlo" has consistency strength squashed between the inaccessible hierarchy and strongly Mahlo cardinals. Ord is Mahlo has interesting consistency strength, as we've mentioned. Say a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise, other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal. Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to persistently force a statement that isn't true to be true. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo. f3f53d0255c3615f4f981a6a9749f68b5c405290 0 107 251 2023-08-30T13:34:28Z RhubarbJayde 25 Created page with "The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordin..." wikitext text/x-wiki The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \( \Delta^0_0 \)-formulae. * The proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \( \Delta^0_0 \)-formulae, with Weak König's Lemma adjoined. 1aba2550dc9dba17502994e1e499269d4742f22a 254 251 2023-08-30T13:38:57Z RhubarbJayde 25 wikitext text/x-wiki The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \( \Delta^0_0 \)-formulae. * The proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \( \Delta^0_0 \)-formulae, with Weak König's Lemma adjoined. * The proof-theoretic ordinal of Peano arithmetic, with induction restricted to \( \Sigma^0_1 \)-formulae. * The proof-theoretic ordinal of primitive recursive arithmetic. d779d213ab0e86624811d86f580b1977becdc7d0 0 17 252 197 2023-08-30T13:36:13Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is likely not significant enough to have its own page.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 69907d3cd361995f209d70abc84146061cfc2ffa 253 252 2023-08-30T13:37:02Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ fd1eab35fefc55beb01422bacce25ea6a70fbc31 0 52 255 123 2023-08-30T13:40:17Z RhubarbJayde 25 wikitext text/x-wiki The ordinal \( \omega^2 \) is the least ordinal which is a limit of limit ordinals, as well as the second infinite additive principal ordinal. It is also equal to the proof-theoretic ordinal of rudimentary function arithmetic, and of Peano arithmetic with induction restricted to \( \Delta^0_0 \)-formulae. It is also the approximate growth rate of Conway's chained arrows in the fast-growing hierarchy. 05787a0f2c0150c3a2922f0399748d126ecd4686 0 108 256 2023-08-30T13:44:30Z RhubarbJayde 25 Redirected page to [[Gap ordinal]] wikitext text/x-wiki #REDIRECT [[Gap ordinal]] a5b406600e3f1f75de2e4cb6c566cce68037a3a3 0 73 257 198 2023-08-30T14:00:39Z RhubarbJayde 25 wikitext text/x-wiki A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).<ref name="MarekSrebrny73" />^p.368 Gap ordinals are very large. This is because, if \( \alpha \) is a gap ordinal, then \( L_\alpha \cap \mathcal{P}(\omega) \) satisfies second-order arithmetic, despite not containing ''all'' subsets of \( \omega \). Therefore, if \( \alpha \) is a gap ordinal, it is admissible, recursively inaccessible, recursively Mahlo, nonprojectible, and more. However, there can still be countable gap ordinals. There is a nice analogy between gap ordinals and cardinals. Note that \( \alpha \) is a cardinal if, for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \). If \( \alpha \) is infinite, we have \( \pi \subseteq \gamma \times \alpha \subseteq \alpha \times \alpha \subseteq V_\alpha^2 \subseteq V_\alpha \) and thus \( \pi \in V_{\alpha + 1} \). Thus, "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in V_{\alpha + 1} \)" is equivalent to being a cardinal. Meanwhile, the least ordinal satisfying the similar but weaker condition "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in L_{\alpha + 1} \)" is equal to the least gap ordinal, since it's equivalent to \( L_\alpha \) satisfying separation.<ref>R.Björn Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, Volume 4, Issue 3, 1972, Pages 229-308</ref> ==Longer gaps== Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha) \cap \mathcal{P}(\omega) = \emptyset\).<ref name="MarekSrebrny73" />^p.365 If such an \( \alpha \) starts a gap, then it is said to start a gap of length \( \gamma \). It is possible for \( \alpha \) to start a gap of length \( > \alpha \): for example, the least \( \alpha \) so that \( \alpha \) starts a gap of length \( \alpha^+ \) is equal to the least admissible which is not locally countable. There can also be second-order gap, and more. An ordinal \( \alpha \) is said to start an \( \eta \)th-order gap of length \( \gamma \) if \( (L_{\alpha+\gamma} \setminus L_\alpha) \cap \mathcal{P}^\eta(\omega) = \emptyset \) and, for all \( \beta < \alpha \), \((L_\alpha \setminus L_\beta) \cap \mathcal {P}^\eta(\omega) \neq \emptyset\). The least ordinal which starts a second-order gap is greater than the least \( \alpha \) which starts a first-order gap of length \( \alpha \), and more. If \( 0^\sharp \) exists, then, for any countable \( \eta\) and any \( \gamma \) at all, there is a countable \( \delta \) which starts an \( \eta \)th-order gap of length \( \gamma \). Meanwhile, if \( V = L \), then there is no countable ordinal starting a first-order gap of length \( \omega_1 \). 02a8cc2cc7bef39e39363cfdf8bde13ed5a7abed 0 109 258 2023-08-30T14:01:41Z RhubarbJayde 25 Redirected page to [[Buchholz's psi-functions]] wikitext text/x-wiki #REDIRECT [[Buchholz's psi-functions]] d16feb993ff3c3edd2198b78aa1ec52158980509 0 110 259 2023-08-30T14:02:06Z RhubarbJayde 25 Redirected page to [[Buchholz's psi-functions]] wikitext text/x-wiki #REDIRECT [[Buchholz's psi-functions]] d16feb993ff3c3edd2198b78aa1ec52158980509 0 111 260 2023-08-30T14:02:35Z RhubarbJayde 25 Redirected page to [[Large cardinal]] wikitext text/x-wiki #REDIRECT [[Large cardinal]] 2d60dc31ecbcde787cd38c02d2a7b0ad2ca93248 0 112 261 2023-08-30T14:44:58Z RhubarbJayde 25 Created page with "An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of Stron..." wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that all countable ordinals reachable from ordinals below \( \max(1, \rho) \) and \( \Omega \) via addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \) are less than \( \rho \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \). The same property applies to [[Buchholz's psi-functions]], Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]], and Rathjen's OCF collapsing a [[weakly compact cardinal]]. == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \) & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 27a4904216713c2777bdb9c98e553e3c9149bbaf 262 261 2023-08-30T14:45:15Z RhubarbJayde 25 wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that all countable ordinals reachable from ordinals below \( \max(1, \rho) \) and \( \Omega \) via addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \) are less than \( \rho \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \). The same property applies to [[Buchholz's psi-functions]], Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]], and Rathjen's OCF collapsing a [[weakly compact cardinal]]. == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \) & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 721f97f7fb58bcd415d440c928252e9080ca5c95 263 262 2023-08-30T14:45:44Z RhubarbJayde 25 /* History */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that all countable ordinals reachable from ordinals below \( \max(1, \rho) \) and \( \Omega \) via addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \) are less than \( \rho \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \). The same property applies to [[Buchholz's psi-functions]], Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]], and Rathjen's OCF collapsing a [[weakly compact cardinal]]. == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \) & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal cf57d129491f7a49a8fb421a4fd6e1beb9b7fbb1 264 263 2023-08-30T14:48:05Z RhubarbJayde 25 wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that all countable ordinals reachable from ordinals below \( \max(1, \rho) \) and \( \Omega \) via addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \) are less than \( \rho \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \) & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 05933ffce0bfd88f62c07701d07f09e8305a697d 0 113 265 2023-08-30T14:53:39Z RhubarbJayde 25 Redirected page to [[Ordinal collapsing function]] wikitext text/x-wiki #REDIRECT [[Ordinal collapsing function]] 747f329a67b39510b5a80a72b1d0cb75a18c6ce3 0 54 266 135 2023-08-30T15:08:16Z RhubarbJayde 25 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatornname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatornname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). This admits an ordinal notation too, as well as a canonical set of fundamental sequences. ca59cff75ce2b868e87b63c5dcfb0a5788dc2eeb 267 266 2023-08-30T15:09:35Z RhubarbJayde 25 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). This admits an ordinal notation too, as well as a canonical set of fundamental sequences. a8295c0cf2caf13daa55394e07f5b5decbc61df7 268 267 2023-08-30T15:11:01Z RhubarbJayde 25 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. 29155b02a3b988a952b8d43b252c3f0bc5b0d28c 0 114 269 2023-08-30T15:34:01Z RhubarbJayde 25 Created page with "An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma). The least additively principal ordinal is 1 since \(0 + 0 = 1\), and all additively principal ordinals other than 1 are limit ordinals. In particular, you can see from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation) that the additively principal ordinals are precisely the ord..." wikitext text/x-wiki An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma). The least additively principal ordinal is 1 since \(0 + 0 = 1\), and all additively principal ordinals other than 1 are limit ordinals. In particular, you can see from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation) that the additively principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some \(\gamma\). As such, the second infinite additively principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additively principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of those is \(\omega^{\omega^2}\). <nowiki>Additively principal ordinals can be generalized to multiplicatively principal ordinals and exponentially principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicatively principal ordinals are to additively principal ordinals as additively principal ordinals are to limit ordinals. However, exponentially principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just the same as the </nowiki>[[epsilon numbers]]. 37eec9ec54a633060131a26e2853b4ec48ba497f 270 269 2023-08-30T15:34:13Z RhubarbJayde 25 wikitext text/x-wiki An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additively principal ordinal is 1 since \(0 + 0 = 1\), and all additively principal ordinals other than 1 are limit ordinals. In particular, you can see from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation) that the additively principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some \(\gamma\). As such, the second infinite additively principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additively principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of those is \(\omega^{\omega^2}\). <nowiki>Additively principal ordinals can be generalized to multiplicatively principal ordinals and exponentially principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicatively principal ordinals are to additively principal ordinals as additively principal ordinals are to limit ordinals. However, exponentially principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just the same as the </nowiki>[[epsilon numbers]]. 1189d3036e1900e8f66fe915f5ff61c3eea7d60a 0 115 271 2023-08-30T16:25:15Z RhubarbJayde 25 Created page with "Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail o..." wikitext text/x-wiki Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail of ordinal analysis, and many believe it can be done using [[Bashicu matrix system|BMS]]. One of the primary interests regarding \(Z_2\) is the study of its subsystems, rather than the whole. This is part of a program called reverse mathematics. Since rational numbers, real numbers, complex numbers, continuous functions on the reals, countable groups, and more can be defined in the language of second-order arithmetic, it turns out many classical theorems in number theory, real analysis, topology, abstract algebra and group theory are provable in \(Z_2\), and most even in weak subsystems! The "big five" are the following:<ref>Subsystems of Second Order Arithmetic, Simpson, S.G., ''Perspectives in Logic'', 2009, ''Cambridge University Press''</ref> * \(\mathrm{RCA}_0\): recursive comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^0_1\)-formulae and induction restricted to \(\Sigma^0_1\)-formulae. \(\mathrm{RCA}_0\) has proof-theoretic ordinal [[Omega^omega|\(\omega^\omega\)]], and it can prove the following famous results: the Baire category theorem, the intermediate value theorem, the soundness theorem, the existence of an algebraic closure of a countable field, the existence of a unique real closure of a countable ordered field. * \(\mathrm{WKL}_0\): weak König's lemma, i.e. \(\mathrm{RCA}_0\) with the additional axiom "every infinite binary tree has an infinite branch" \(\mathrm{WKL}_0\) has the same proof-theoretic ordinal as \(\mathrm{RCA}_0\), but is able to prove some non-induction related theorems which \(\mathrm{RCA}_0\) can't, such as: the Heine/Borel covering lemma, every continuous real-valued function on [0, 1], or even any compact metric space, is bounded, the local existence theorem for solutions of (finite systems of) ordinary differential equations, Gödel’s completeness theorem, every countable commutative ring has a prime ideal and Brouwer’s fixed point theorem. * \(\mathrm{ACA}_0\): arithmetical comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^1_0\)-formulae \(\mathrm{ACA}_0\) and Peano arithmetic have the same first-order consequences and thus the same proof-theoretic ordinal: namely, [[Epsilon numbers|\(\varepsilon_0\)]]. Not much has been said regarding \(\mathrm{ACA}_0\)'s ordinary, non-number-theoretical consequences. * \(\mathrm{ATR}_0\): arithmetical transfinite recursion, i.e. \(\mathrm{ACA}_0\) with the additional axiom "every arithmetical operator can be iterated along any countable well-ordering" \(\mathrm{ATR}_0\) has the proof-theoretic ordinal [[Feferman-Schütte ordinal|\(\Gamma_0\)]], which is part of the reason why the ordinal in question is claimed to be the limit of what can be predicatively defined. \(\mathrm{ATR}_0\) can prove the following: any two countable well orderings are comparable, any two countable reduced Abelian p-groups which have the same Ulm invariants are isomorphic, and that every uncountable closed, or analytic, set has a perfect subset. * \(\Pi^1_1 \mathrm{-CA}_0\): \(\Pi^1_1\)-comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Pi^1_1\)-formulae \(\Pi^1_1 \mathrm{-CA}_0\) has a significantly higher proof-theoretic ordinal than the previous entries - namely, [[Buchholz ordinal|\(\psi_0(\Omega_\omega)\)]]. It can prove the following: every countable Abelian group is the direct sum of a divisible group and a reduced group, the Cantor/Bendixson theorem, a set is Borel iff it and its complement are analytic, any two disjoint analytic sets can be separated by a Borel set, coanalytic uniformization, and more. There are also even stronger systems such as \(\Pi^1_1 \mathrm{-TR}_0\), which is \(\Pi^1_1 \mathrm{-CA}_0\) with the axiom "every \(\Pi^1_1\)-definable operator can be iterated along any countable well-ordering", \(\Pi^1_2 \mathrm{-CA}_0\), and more. The former has proof-theoretic ordinal [[Extended Buchholz ordinal|EBO]], while the latter's proof-theoretic ordinal hasn't been precisely calibrated but has been bound.<ref>Determinacy and \(\Pi^1_1\) transfinite recursion along \(\omega\), Takako Nemoto, 2011</ref><ref>An ordinal analysis of \(\Pi_1\)-Collection, Toshiyasu Arai, 2023</ref> 04d73e9fa4b6ef1e4cb1110fd8f103f6a86f04f7 0 107 272 254 2023-08-30T16:26:52Z RhubarbJayde 25 wikitext text/x-wiki The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordinal of [[Second-order arithmetic|\(\mathrm{RCA}_0\)]] * The proof-theoretic ordinal of [[Second-order arithmetic|\(\mathrm{WKL}_0\)]]. * The proof-theoretic ordinal of Peano arithmetic, with induction restricted to \( \Sigma^0_1 \)-formulae. * The proof-theoretic ordinal of primitive recursive arithmetic. 0cc5b21913fd6add9342257e6ac9949204c9f091 0 96 273 235 2023-08-30T16:28:06Z RhubarbJayde 25 wikitext text/x-wiki The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). 6fdf3c2063765109d592c021154d4e14a5fbcd92 300 273 2023-08-30T20:33:59Z RhubarbJayde 25 wikitext text/x-wiki The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's [[Buchholz's psi-functions|original set]] of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). ee0e11b916b51a4e95f8d32f0f3d16da7e2cf650 0 116 274 2023-08-30T16:28:43Z RhubarbJayde 25 Redirected page to [[Additive principal ordinals]] wikitext text/x-wiki #REDIRECT [[Additive principal ordinals]] 067aac591bdbd1143d83fc582434b15dcd51ed6d 0 16 275 115 2023-08-30T16:30:33Z RhubarbJayde 25 wikitext text/x-wiki A normal function is an [[ordinal function]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]. Veblen's fixed point lemma, which is essential for constructing the [[Veblen hierarchy]], guarantees that, not only does every normal function have a fixed point, but the class of fixed points is unbounded and their enumeration function is also normal. ddff141cd6f3f79853f41654968524dd3162e1e9 0 117 276 2023-08-30T16:31:12Z RhubarbJayde 25 Redirected page to [[Ordinal#Ordinal arithmetic]] wikitext text/x-wiki #REDIRECT [[Ordinal#Ordinal%20arithmetic]] c8c1dd8ac2cb078bb39962d27a56442872a4a8be 0 49 277 117 2023-08-30T16:32:38Z RhubarbJayde 25 wikitext text/x-wiki The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes). 539c7d8bf0fe09ba0aaf02d79f5f80fe515e0f8c 0 118 278 2023-08-30T16:33:06Z RhubarbJayde 25 Redirected page to [[Ordinal#Von Neumann definition]] wikitext text/x-wiki #REDIRECT [[Ordinal#Von%20Neumann%20definition]] 6a1c9efadf16b23f223431cb330d375fcfb0e893 0 119 279 2023-08-30T16:33:23Z RhubarbJayde 25 Redirected page to [[Inaccessible cardinal]] wikitext text/x-wiki #REDIRECT [[Inaccessible cardinal]] 7388e37fce8544c0ea026f61f8c4ebb0f8d6a9cd 0 88 280 226 2023-08-30T16:33:57Z RhubarbJayde 25 wikitext text/x-wiki The Buchholz ordinal is the limit of Wilfried Buchholz's original set of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), [[Bachmann-Howard ordinal|\( \mathrm{BHO} \)]], \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, i.e. [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]], or of Peano arithmetic with finitely iterated inductive definitions, as well as the limit of pair sequence system in [[Bashicu matrix system]]. It can be viewed as an extension of the Bachmann-Howard ordinal by allowing higher uncountable cardinals, which iteratively collapse above \( \Omega \). 17b9953a762cf0fd3b4b788a18b03ac1b2f12593 299 280 2023-08-30T20:33:37Z RhubarbJayde 25 wikitext text/x-wiki The Buchholz ordinal is the limit of Wilfried Buchholz's [[Buchholz's psi-functions|original set]] of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), [[Bachmann-Howard ordinal|\( \mathrm{BHO} \)]], \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, i.e. [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]], or of Peano arithmetic with finitely iterated inductive definitions, as well as the limit of pair sequence system in [[Bashicu matrix system]]. It can be viewed as an extension of the Bachmann-Howard ordinal by allowing higher uncountable cardinals, which iteratively collapse above \( \Omega \). 582d885de73ff3fb10bbc9b4bafeae616a7899a9 0 120 281 2023-08-30T16:52:12Z RhubarbJayde 25 Created page with "Kripke-Platek set theory, commonly abbreviated KP, is a weak foundation of set theory used to define admissible ordinals, which are immensely important in ordinal analysis and \(\alpha\)-recursion theory. In terms of proof-theoretic strength, its proof-theoretic ordinal is the [[Bachmann-Howard ordinal|BHO]], and it is thus intermediate between [[Second-order arithmetic|\(\mathrm{ATR}_0\)]] and [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]]. The axioms of KP are..." wikitext text/x-wiki Kripke-Platek set theory, commonly abbreviated KP, is a weak foundation of set theory used to define admissible ordinals, which are immensely important in ordinal analysis and \(\alpha\)-recursion theory. In terms of proof-theoretic strength, its proof-theoretic ordinal is the [[Bachmann-Howard ordinal|BHO]], and it is thus intermediate between [[Second-order arithmetic|\(\mathrm{ATR}_0\)]] and [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]]. The axioms of KP are the following: * The axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of induction: transfinite induction along the \(\in\)-relation (this implies the axiom of foundation) * Axiom of empty set: There exists a set with no members. * Axiom of pairing: If x, y are sets, then so is {x, y}. * Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x. * Axiom of infinity: there is an inductive set. * Axiom of \(\Delta_0\)-separation: Given any set \(X\) and any \(\Delta_0\)-formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of \(\Delta_0\)-collection: If \(\varphi(x,y)\) is a \(\Delta_0\)-formula so that \(\forall x \exists y \varphi(x,y)\), then for all \(X\), there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). These axioms lead to close connections between KP, \(\alpha\)-recursion theory, and the theory of admissible ordinals. A set \(M\) is admissible if, and only if, it satisfies KP. We say \(\alpha\) is admissible if \(L_\alpha\) is admissible. You can see that this holds if and only if \(\alpha > \omega\) is a limit ordinal and, for every \(\Delta_0(L_\alpha)\)-definable \(f: L_\alpha \to L_\alpha\) and \(x \in L_\alpha\), \(f<nowiki>''</nowiki>x \in L_\alpha\) as well. The least admissible ordinal is the [[Church-Kleene ordinal]]. Note that some authors drop the axiom of infinity and consider \(\omega\) an admissible ordinal too. Note that \(\Delta_0\)-collection actually implies \(\Sigma_1\)-collection. Kripke-Platek set theory may be imagined as being the minimal system of set theory which has infinite sets and allows one to do "computable" and "predicative" definitions over sets. This is because the \(\Sigma_1(L_\alpha)\)-definable functions are considered analogues of computable ones, and, in the context of separation, \(\Delta_0\)-formulae are considered predicative or primitive due to not referencing the totality of the universe. ed4f095abc3e3229a0c7d839737967030a9a8469 0 121 282 2023-08-30T17:15:09Z RhubarbJayde 25 Created page with "The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of..." wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals is a cumulative hierarchy if, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. ddccc76dc761f9115ae6bf626e1ebf32eb74e990 285 282 2023-08-30T17:37:24Z RhubarbJayde 25 wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals is a cumulative hierarchy if, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Alternate meaning == An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals. Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. 47dd1d96e61813ee33f4900f2c127b424ff96a38 287 285 2023-08-30T17:58:21Z RhubarbJayde 25 /* Alternate meaning */ wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals is a cumulative hierarchy if, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Alternate meaning == An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals. Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U}\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U}\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U}\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi^0_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. f95dfa074313ec459569835bfc57f17bbb7fdb8e 288 287 2023-08-30T17:58:46Z RhubarbJayde 25 /* Alternate meaning */ wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals is a cumulative hierarchy if, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Alternate meaning == An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals. Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U})\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi^0_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. 3ae47ddb4157ead4369f2606267af73a8e005a3f 289 288 2023-08-30T17:59:04Z RhubarbJayde 25 /* Alternate meaning */ wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals is a cumulative hierarchy if, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Alternate meaning == An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals. Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi^0_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. 5bd4e07e86f81c9bcc6f9a2e3aaa600129781d69 290 289 2023-08-30T18:00:15Z RhubarbJayde 25 /* Alternate meaning */ wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals is a cumulative hierarchy if, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Alternate meaning == An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals. Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecti#ng ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. de617a1fcd27d0cbdc80eaeca1225527036ff55b 0 122 283 2023-08-30T17:15:40Z RhubarbJayde 25 Redirected page to [[Reflection principle]] wikitext text/x-wiki #REDIRECT [[Reflection principle]] 54b8fc734093b182a39853b2370bd9099aedb6b1 0 123 284 2023-08-30T17:16:08Z RhubarbJayde 25 Redirected page to [[Reflection principle]] wikitext text/x-wiki #REDIRECT [[Reflection principle]] 54b8fc734093b182a39853b2370bd9099aedb6b1 0 106 286 250 2023-08-30T17:55:29Z RhubarbJayde 25 wikitext text/x-wiki A Mahlo cardinal is a certain type of [[large cardinal]] used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less popular in the literature, weakly Mahlo cardinals are more popular than strongly Mahlo cardinals in apeirological circles, but less popular in the literature. This is essentially due to the fact that weakly Mahlo cardinals are defined in terms of weakly inaccessibles, and strongly Mahlo cardinals are defined in terms of strongly inaccessibles. Essentially, we say a cardinal \( \kappa \) is weakly Mahlo if every club \( C \subseteq \kappa \) contains a regular cardinal. First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{\lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars. Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get \( (+1) \)-stable ordinals, and removing it altogether grants you a condition equivalent to regularity! Then, being Mahlo is obtained by adding the condition that such a closure point must be regular: in other words, \( \kappa \) is Mahlo iff. for any function \( f: \kappa \to \kappa \), there is a regular closure point of \( f \) below \( \kappa \). You can see that this is equivalent to Mahloness by noting that the set of closure points of any function is club, and any club is equal to the set of closure points of some function. Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy. == Strongly Mahlo == Strong Mahloness is obtained by replacing "contains a regular cardinal" with "contains a strongly inaccessible cardinal". Clearly any strongly Mahlo cardinal is weakly Mahlo, since every strongly inaccessible cardinal is regular, and the results above can be generalized to show any strongly Mahlo cardinal is strongly hyper-inaccessible. Like the situation between weakly and strongly inaccessible cardinals, \( \mathrm{GCH} \) implies weakly and strongly Mahlo cardinals are the same, while other axioms imply that \( 2^{\aleph_0} \) can be weakly Mahlo. == Ord is Mahlo == "Ord is Mahlo" is an assertion that, as one can likely guess, asserts that every function \( f: \mathrm{Ord} \to \mathrm{Ord} \) has a strongly inaccessible closure point. Clearly, "Ord is Mahlo" implies that there is a proper class of inaccessible cardinals, 1-inaccessible cardinals, and more. However, if \( \kappa \) is Mahlo, then \( V_\kappa \) satisfies "Ord is Mahlo", and thus "Ord is Mahlo" has consistency strength squashed between the inaccessible hierarchy and strongly Mahlo cardinals. Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from [[Reflection principle|here]] that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength. Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to persistently force a statement that isn't true to be true. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo. de5571e2776454bd0584d31c78a9728e104b1617 0 74 291 209 2023-08-30T19:44:47Z EricABQ 5 added algebraic properties wikitext text/x-wiki The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes [[zero]]. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite [[ordinal]]s. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. In many areas such as geometry or number theory, they are taken as primitives and not formally treated ===Von Neumann ordinals=== In [[ZFC]] and other set theories without urelements, we can define natural numbers by applying the definition of Von Neumann ordinals to finite ordinals. In this view, each natural number is the set of previous naturals: \( 0 = \varnothing \) and \( n + 1 = \{0, \dots, n\} \). ===Zermelo ordinals=== [[:wikipedia:Ernst Zermelo|Ernst Zermelo]] provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). Unlike the Von Neumann ordinals, Zermelo's encoding can only be used to represent finite ordinals. ===Frege and Russell=== During the early development of foundational philosophy and logicism, [[:wikipedia:Gottlob Frege|Gottlob Frege]] and [[:wikipedia:Bertrand Russell|Bertrand Russell]] proposed defining a natural number \( n \) as the equivalence [[class]] of all sets with [[cardinality]] \( n \). This definition cannot be realized in ZFC, because the classes involved are [[proper class]]es, except for \( n = 0 \). ===Church numerals=== In the [[lambda calculus]], the standard way to encode natural numbers is as Church numerals, developed by [[:wikipedia:Alonzo Church|Alonzo Church]]. In this encoding, each natural number \( n \) is identified with a function that returns the composition of its input with itself \( n \) times: \( 0 := \lambda f. \lambda x. x \), \( 1 := \lambda f. \lambda x. f x \), \( 2 := \lambda f. \lambda x. f (f x) \), etc. ==Theories of arithmetic== Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]]. ==Algebraic properties of the natural numbers== The natural numbers (including zero) are closed under addition and multiplication. They satisfy commutativity and associativity of both operations, and distributivity of multiplication over addition. They form a monoid under addition, which is the free monoid with one generator. In addition, positive naturals form a monoid under multiplication -- the free monoid with countably infinite generators, which are the prime numbers. 3c4ad3b86c0588d1d1e781666cd18b61bea55839 0 100 292 240 2023-08-30T19:51:56Z 24.43.123.81 0 wikitext text/x-wiki The Bird ordinal (sometimes called Bird's ordinal) is an intermediate ordinal between the [[Takeuti-Feferman-Buchholz ordinal]] and [[Extended Buchholz ordinal]] which occurs occasionally in apeirological notations such as [[Bashicu matrix system|BMS]]. It was named by the apeirological community in honor of Chris Bird. This is because it is believed to correspond to the limit of his final system of array notations, and thus the growth rate of a natural extension of his U function. It can be written as \( \psi_0(\Omega_\Omega) \) (not to be confused with [[Buchholz ordinal|\( \psi_0(\Omega_\omega) \)]]) in Denis Maksudov's extension of Wilfried Buchholz's system of ordinal collapsing functions. It is believed to correspond to the proof-theoretic ordinal of \( \mathrm{Aut}(\mathrm{ID}) \), the minimal extension of Peano arithmetic so that, if it proves transfinite induction along a recursive well-order of order-type \( \alpha \), then it also is able to deal with iterated inductive definitions of length \( \alpha \). This makes it essentially the maximal possible extension of the notion of iterated inductive definitions, without the addition of second-order schemata, and shows is much greater than the [[Takeuti-Feferman-Buchholz ordinal]], which is only able to deal with iterated inductive definitions of length \( \omega \). One could possibly consider the analogy that the Bird ordinal is to the Buchholz ordinal as \( \varphi(2,0,0) \) is to [[Feferman-Schütte ordinal|\( \Gamma_0 = \varphi(1,0,0) \)]], although this could potentially be seen as underestimating the size of the Bird ordinal. Alternatively, using the correspondence between identical expressions in [[extended Buchholz's function]] and [[nothing OCF]], the Bird ordinal can be seen as an analog of \( \Gamma_0 \) while the Buchholz ordinal is analogous to \( \varepsilon_0 \). fd24bbdd4b8c464cb5b7e6963a836b0a7758b71e 0 17 293 253 2023-08-30T19:55:54Z 24.43.123.81 0 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC, but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 8128e2c269d9ee97cc463ef9698235f689c273dd 306 293 2023-08-30T20:39:02Z RhubarbJayde 25 wikitext text/x-wiki == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ d7eec0f63ce458edd6e47002bcc324ab206e5b04 2 13 294 27 2023-08-30T20:19:07Z Augigogigi 2 wikitext text/x-wiki sbocf == Definition == * \( T_{a}(b) = \) * \( \tau(a) = \) * \( \kappa(0) = \) == Fundamental Sequences == bla bla bla function: <div style="width:40%;"> \begin{align} [] : \text{S} \times \mathbb{N} \rightarrow& \text{S} \\ (\text{S},n) \mapsto& \text{S}[n] \end{align} </div> bla bla bla bla bla == Analysis == {| class="wikitable" |- ! Expression !! Shorthand |- | \( \tau(a) \) || \( \Omega_{a} \) |- | \( \tau(B+a) \) || \( M_{a} \) |- | \( \tau(B\cdot2+a) \) || \( N_{a} \) |- | \( \tau(B^{2}+a) \) || \( G_{a} \) |} {| class="wikitable" ! SbOCF !! Ordinal !! Xi !! BMS |- | \( T_{\tau(0)}(0) \) || \( 1 \) || - || \( (0) \) |- | \( T_{\tau(0)}(1) \) || \( \omega \) || - || \( (0)(1) \) |- | \( T_{\tau(0)}(2) \) || \( \omega^{2} \) || - || \( (0)(1)(1) \) |- | \( T_{\tau(0)}(T_{\tau(0)}(0)) \) || \( \omega^{\omega} \) || - || \( (0)(1)(2) \) |- | \( T_{\tau(0)}(T_{\tau(0)}(T_{\tau(0)}(0))) \) || \( \omega^{\omega^{\omega}} \) || - || \( (0)(1)(2)(3) \) |- | \( T_{\tau(0)}(\tau(0)) \) || \( \varepsilon_{0} \) || - || \( (0,0)(1,1) \) |- | \( T_{\tau(0)}(\tau(0)+1) \) || \( \varepsilon_{0}\cdot\omega \) || - || \( (0,0)(1,1)(1,0) \) |- | \( T_{\tau(0)}(\tau(0)+T_{\tau(0)}(0)) \) || \( \varepsilon_{0}\cdot\omega^{\omega} \) || - || \( (0,0)(1,1)(1,0)(2,0) \) |- | \( T_{\tau(0)}(\tau(0)+\tau(0)) \) || \( \varepsilon_{0}^{2} \) || - || \( (0,0)(1,1)(1,0)(2,1) \) |- | \( T_{\tau(0)}(\tau(0)\cdot\omega) \) || \( \varepsilon_{0}^{\omega} \) || - || \( (0,0)(1,1)(1,1) \) |} 82c9e2dcd06ca6519bf8c32a1c3f079c489d725e 0 103 295 244 2023-08-30T20:24:52Z RhubarbJayde 25 wikitext text/x-wiki <nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "recursive ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning there must be some ordinals which are still countable but they aren't recursive - i.e: they're so large that all well-orders they code are so complex that they are uncomputable. The least such is the Church-Kleene ordinal. Note that there is still a well-order on the natural numbers with order type \( \omega_1^{\mathrm{CK}} \) that is computable with an </nowiki>[[Infinite time Turing machine|''infinite time'' Turing machine]], since they are able to solve the halting problem for ordinary Turing machines and thus diagonalize over the recursive ordinals. Also, note that given computable well-orders on the natural numbers with order types \( \alpha \) and \( \beta \), it is possible to construct computable well-orders with order-types \( \alpha + \beta \), \( \alpha \cdot \beta \) and \( \alpha^{\beta} \) and much more, meaning that the Church-Kleene ordinal is not pathological and in fact a limit ordinal, [[Epsilon numbers|epsilon number]], [[Strongly critical ordinal|strongly critical]], and more. <nowiki>It has a variety of other convenient definitions. One of them has to do with the constructible hierarchy - \( \omega_1^{\mathrm{CK}} \) is the least </nowiki>[[admissible]] ordinal. In other words, it is the least limit ordinal \( \alpha > \omega \) so that, for any \( \Delta_0(L_\alpha) \)-definable function \( f: L_\alpha \to L_\alpha \), then, for all \( x \in L_\alpha \), \( f<nowiki>''</nowiki>x \in L_\alpha \). That is, the set of constructible sets with rank at most \( \omega_1^{\mathrm{CK}} \) is closed under taking preimages of an infinitary analogue of the primitive recursive functions. Note that this property still holds for \( \Sigma_1(L_\alpha) \)-functions, an infinitary analogue of Turing-computable functions, which makes sense, since the ordinals below \( \omega_1^{\mathrm{CK}} \) are a very robust class and closed under computable-esque functions. It is, in particular, equivalent to the statement: for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point (equivalent to a fixed point) of \( f \) below \( \alpha \). This also is a good explanation, since it shows it's a limit of epsilon numbers (and thus itself an epsilon number), of strongly critical numbers, etc. and why it's greater than recursive ordinals like the [[Bird ordinal]]. However, the case with \( \Sigma_2(L_\alpha) \)-functions produces a much stronger notion known as \( \Sigma_2 \)-admissibility. <nowiki>Note that, like how there is a fine hierarchy of recursive ordinals and functions on them, there is a fine hierarchy of nonrecursive ordinals, above \( \omega_1^{\mathrm{CK}} \), arguably richer.</nowiki> <nowiki>One thing to note is that many known ordinal collapsing functions are, or should be, \( \Sigma_1(L_{\omega_1^{\mathrm{CK}}}) \)-definable. Thus, the countable collapse is actually a recursive collapse, and replacing \( \Omega \) with \( \omega_1^{\mathrm{CK}} \) in an ordinal collapsing function is a possibility. While many authors do this, since it allows them to use structure more efficiently and not assume </nowiki>[[Large cardinal|large cardinal axioms]], more cumbersome proofs would be necessary, and this has led many authors such as Rathjen to instead opt for the traditional options, or use uncountable intermediates between countable nonrecursive fine structure and large cardinals, such as the reducibility hierarchy. 87090ff88e1fd47ae4e88e6d4492ed54cecc0900 0 57 296 145 2023-08-30T20:28:32Z RhubarbJayde 25 wikitext text/x-wiki A set \(M\) is admissible if \((M,\in)\) is a model of [[Kripke-Platek set theory]]. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\), which is itself equivalent to \(\alpha > \omega\), \(\alpha\) being a limit ordinal and \(L_\alpha\) being closed under preimages of \(\alpha\)-recursively enumerable functions.<ref>Probably in Barwise somewhere</ref> The least admissible ordinal is [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]], although some authors omit the axiom of infinity from KP and consider [[Omega|\(\omega\)]] to be admissible. d198a04831ee796be80a59589c5dc3c82394c51d 0 55 297 146 2023-08-30T20:29:28Z RhubarbJayde 25 Redirected page to [[Uncountable]] wikitext text/x-wiki #REDIRECT [[Uncountable]] 7e5045abe59043d5b5c6d85b79b4f5e0b8fbf5c0 0 86 298 224 2023-08-30T20:33:18Z RhubarbJayde 25 wikitext text/x-wiki The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). Buchholz further extended this to [[Buchholz's psi-functions|his famous set]] of collapsing functions, whose limit is the much larger [[Buchholz ordinal]]. 349c33822e78417ea29a95a9114eef8627651952 0 124 301 2023-08-30T20:34:18Z RhubarbJayde 25 Redirected page to [[Bachmann-Howard ordinal]] wikitext text/x-wiki #REDIRECT [[Bachmann-Howard ordinal]] 03d2a8c79515ff9f4907a83c444918244bbfbbab 0 125 302 2023-08-30T20:35:26Z RhubarbJayde 25 Redirected page to [[Large Veblen ordinal]] wikitext text/x-wiki #REDIRECT [[Large Veblen ordinal]] d0dc149bdfe38ef71c4d89c9c2ecdf8b6aabe547 0 126 303 2023-08-30T20:36:13Z RhubarbJayde 25 Redirected page to [[Takeuti-Feferman-Buchholz ordinal]] wikitext text/x-wiki #REDIRECT [[Takeuti-Feferman-Buchholz ordinal]] c8fde5c61d7418ac5f17bda4d6191e7d77d0a339 0 127 304 2023-08-30T20:36:30Z RhubarbJayde 25 Redirected page to [[Extended Buchholz ordinal]] wikitext text/x-wiki #REDIRECT [[Extended Buchholz ordinal]] 38daf2b7770537c4bfd2009af817c56b4b976a1a 0 98 305 238 2023-08-30T20:37:08Z RhubarbJayde 25 wikitext text/x-wiki The Extended Buchholz ordinal, sometimes known as OFP (short for omega-fixed-point), is the limit of an extension of Buchholz's original set of ordinal collapsing functions, defined by Denis Maksudov, which allows to collapse ordinals such as \( \Omega_{\omega + 1} \) (which corresponds to the [[Takeuti-Feferman-Buchholz ordinal]]), \( \Omega_{\omega^2} \) (which is believed to correspond to the [[Bashicu matrix system|BMS]] matrix (0,0,0)(1,1,1)(2,1,1)), or \( \Omega_{\Omega} \) (which corresponds to the [[Bird ordinal]]). It has not been widely studied in the literature, but is common in amateur apeirological discussions, and is known to correspond to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-TR}_0 \), a second-order strengthening of arithmetical transfinite recursion (the proof theoretic ordinal of which is the [[Feferman-Schütte ordinal]]). Thus, one could claim that it is to the [[Buchholz ordinal]] as the [[Feferman-Schütte ordinal]] is to [[Epsilon numbers|\( \varepsilon_0 \)]], although this may be an understatement. 8605503e1928a662b04bc105946fc10257210f35 0 71 307 194 2023-08-30T20:40:33Z RhubarbJayde 25 wikitext text/x-wiki A weakly compact cardinal is a certain kind of [[large cardinal]]. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indescribable. * \(\kappa\) is \(\kappa\)-unfoldable. * The partition property \(\kappa \to (\kappa)^2_2\) holds. Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem. The existence of a weakly compact cardinal is not provable in ZFC, assuming its existence - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, since "\(\kappa\) is weakly compact" is a \(\Pi^1_2\) property of \(V_\kappa\),<ref>Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:[https://doi.org/10.1007%2F978-3-540-88867-3_2 10.1007/978-3-540-88867-3_2]. ISBN 3-540-00384-3</ref> i.e. a \(\Pi_2\) property of \(V_{\kappa+1}\),<ref>J. D. Hamkins, "[https://jdh.hamkins.org/local-properties-in-set-theory/ Local properties in set theory]" (2014), blog post. Accessed 29 August 2023.</ref> a totally reflecting cardinal, or even a \(\Pi^1_2\)-indescribable cardinal, is larger than the least weakly compact cardinal. Note that, unlike the relation between weakly and strongly inaccessible cardinals, and weakly and strongly Mahlo cardinals, strongly compact cardinals are always significantly greater than weakly compact cardinals, both in terms of consistency strength and size. ==References== <references /> 0cf29603042865383a92dacb1485a1d302c8d169 0 128 308 2023-08-30T20:41:42Z RhubarbJayde 25 Redirected page to [[Gandy ordinal]] wikitext text/x-wiki #REDIRECT [[Gandy ordinal]] b6fb3e3ea0047b87d66914c2a874ab9a2a7808b1 0 129 309 2023-08-30T20:42:30Z RhubarbJayde 25 Redirected page to [[Gandy ordinal]] wikitext text/x-wiki #REDIRECT [[Gandy ordinal]] b6fb3e3ea0047b87d66914c2a874ab9a2a7808b1 0 56 310 230 2023-08-30T20:47:17Z TrialPurpleCube 26 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals beyond \( \Gamma_0 \) can be written using an extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries, to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]]. b1d507abcbfc97bb5a077c29eb4ed8c325ed938b 312 310 2023-08-30T20:58:54Z RhubarbJayde 25 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries. However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient. ecc6c302e01da66127c3c95873ea1a0fa5183ed4 0 130 311 2023-08-30T20:55:19Z RhubarbJayde 25 Created page with "For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An [[ordinal]] \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next [[admissible]] ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy i..." wikitext text/x-wiki For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An [[ordinal]] \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next [[admissible]] ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy is called non-Gandy or, colloquially, bad. The least bad ordinal is equal to the least ordinal which is \(\Sigma^1_1\)-reflecting<ref>R. Gostanian. The Next Admissible Ordinal. Ann. Math. Logic, 17:171–203, 1979</ref>, which is greater than the least \(\alpha\) which is \(\alpha^+\)-stable, and less than the least \(\alpha\) which is \(\alpha^++1\)-stable.<ref>The Order of Reflection, J. P. Aguilera</ref> The structure of stability at the level of bad ordinals and beyond becomes a lot more complex, due to the highly nonlinear nature of iterated \(\Sigma^1_1\)- and \(\Pi^1_1\)-reflection, and one needing more and more iterations of \(\delta\) to reach the next admissible ordinal. It is believed that the least bad ordinal may be useful to act as a "diagonalizer" in a possible formalization in Dmytro Taranovsky's "Degrees of Reflection". f49b0f6f33f63e8f257ec6858c648958680c1c7b 313 311 2023-08-30T21:00:27Z RhubarbJayde 25 wikitext text/x-wiki For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An [[ordinal]] \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next [[admissible]] ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy is called non-Gandy or, colloquially, bad. The least bad ordinal is equal to the least ordinal which is \(\Sigma^1_1\)-reflecting<ref>R. Gostanian. The Next Admissible Ordinal. Ann. Math. Logic, 17:171–203, 1979</ref>, which is greater than the least \(\alpha\) which is \(\alpha^+\)-stable, and less than the least \(\alpha\) which is \(\alpha^++1\)-stable.<ref>The Order of Reflection, J. P. Aguilera</ref> The structure of stability at the level of bad ordinals and beyond becomes a lot more complex, due to the highly nonlinear nature of iterated \(\Sigma^1_1\)- and \(\Pi^1_1\)-reflection, and one needing more and more iterations of \(\delta\) to reach the next admissible ordinal. Proper usage of previously ignored, intricate bad ordinal structure is believed to be essential to maximising the strength of ordinal collapsing functions and associated ordinal notations. For example, it is believed that the least bad ordinal may be useful to act as a "diagonalizer" in a possible formalization in Dmytro Taranovsky's "Degrees of Reflection". 0f02be2ce99e4f71ec2999f5474f2c8e3885bfab 0 15 314 172 2023-08-30T21:01:05Z TrialPurpleCube 26 wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs). Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> 4f28fe0ae2180fe29e3753d9357f4d90cc74d895 0 131 315 2023-08-30T21:09:11Z RhubarbJayde 25 Created page with "Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\b..." wikitext text/x-wiki Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting. The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals, and more, but not yet \(\Pi^1_1\)-reflecting. In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in [[Ordinal collapsing function|OCFs]] for ordinal analyses of systems beyond [[Kripke-Platek set theory]] with full reflection. This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref>Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref> 1110fffc03c34f82680c44df8a57b42775de5958 316 315 2023-08-30T23:36:33Z C7X 9 More citation wikitext text/x-wiki Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting.<ref name="RichterAczel74">Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref>^{Section 6} The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals, and more, but not yet \(\Pi^1_1\)-reflecting. In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in [[Ordinal collapsing function|OCFs]] for ordinal analyses of systems beyond [[Kripke-Platek set theory]] with full reflection. This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref name="RichterAczel74" />^{Section 6} a27d5779a24021d616002cbb2da709e4ada2e705 317 316 2023-08-31T00:10:20Z C7X 9 wikitext text/x-wiki Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting.<ref name="RichterAczel74">Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref>^{Section 6} The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals,<ref>To be proved on this page</ref> and more, but not yet \(\Pi^1_1\)-reflecting.<ref>To be proved on this page</ref> In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in [[Ordinal collapsing function|OCFs]] for ordinal analyses of systems beyond [[Kripke-Platek set theory]] with full reflection. This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref name="RichterAczel74" />^{Section 6} ==(+η)-stability== Theorem: Let \(n<\omega\). If \(\alpha\) is \((+n+1)\)-stable, then \(\alpha\) is \(\Pi^1_0\)-reflecting on the class of \((+n)\)-stable ordinals below \(\alpha\). Proof: Corollary: \(\Pi^1_1\)-reflecting ordinals \(\alpha\) are \(\alpha^+\)-stable, therefore they are \(\alpha+3\)-stable. So each \(\Pi^1_1\)-reflecting ordinal is \(\Pi^1_0\)-reflecting on the \((+2)\)-stable ordinals below it. 9939e309dbf1d14b34f7a29085503b2c96a1c4d2 319 317 2023-08-31T00:29:43Z C7X 9 wikitext text/x-wiki Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting.<ref name="RichterAczel74">Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref>^{Section 6} The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals,<ref>To be proved on this page</ref> and more, but not yet \(\Pi^1_1\)-reflecting.<ref>To be proved on this page</ref> In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in [[Ordinal collapsing function|OCFs]] for ordinal analyses of systems beyond [[Kripke-Platek set theory]] with full reflection. This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref name="RichterAczel74" />^{Section 6} ==(+η)-stability== Theorem: Let \(n<\omega\). If \(\alpha\) is \((+n+1)\)-stable, then \(\alpha\) is \(\Pi^1_0\)-reflecting on the class of \((+n)\)-stable ordinals below \(\alpha\). Proof: Assume \(\alpha\) is (+2)-stable and \(\phi(\vec x)\) is a first-order formula with parameters from \(L_\alpha\) such that \(L_\alpha\vDash\phi(\vec x)\). WIP Corollary: \(\Pi^1_1\)-reflecting ordinals \(\alpha\) are \(\alpha^+\)-stable, therefore they are \(\alpha+3\)-stable. So each \(\Pi^1_1\)-reflecting ordinal is \(\Pi^1_0\)-reflecting on the \((+2)\)-stable ordinals below it. 0e0ddfe128dcf7fe405678ddc2139b8367886150 326 319 2023-08-31T05:57:19Z C7X 9 /* (+η)-stability */ wikitext text/x-wiki Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting.<ref name="RichterAczel74">Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref>^{Section 6} The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals,<ref>To be proved on this page</ref> and more, but not yet \(\Pi^1_1\)-reflecting.<ref>To be proved on this page</ref> In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in [[Ordinal collapsing function|OCFs]] for ordinal analyses of systems beyond [[Kripke-Platek set theory]] with full reflection. This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref name="RichterAczel74" />^{Section 6} ==(+η)-stability== Theorem: Let \(n<\omega\). If \(\alpha\) is \((+n+1)\)-stable, then \(\alpha\) is \(\Pi^1_0\)-reflecting on the class of \((+n)\)-stable ordinals below \(\alpha\). Proof: Assume \(\alpha\) is (+2)-stable and \(\phi(\vec x)\) is a first-order formula with parameters from \(L_\alpha\) such that \(L_\alpha\vDash\phi(\vec x)\). \(L_{\alpha+2}\) satisfies \(\exists\gamma(\phi^{L_\gamma}(\vec x)\land L_\gamma\prec_{\Sigma_1}L_{\gamma+1})\) with \(\alpha\) as a witness of such a \(\gamma\) (is stability well-behaved in \(L_{\gamma+2}\), a successor stage of \(L\)?), and this is a \(\Sigma_1\) formula, so by \(L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\), \(L_\alpha\) satisfies this as well. Then there is a \(\gamma<\alpha\) such that \(\phi^{L_\gamma}(\vec x)\) and \(\gamma\) is \((+1)\)-stable. QED Corollary: \(\Pi^1_1\)-reflecting ordinals \(\alpha\) are \(\alpha^+\)-stable, therefore they are \(\alpha+3\)-stable. So each \(\Pi^1_1\)-reflecting ordinal is \(\Pi^1_0\)-reflecting on the \((+2)\)-stable ordinals below it. 27d24160f8fb20f81cbbb3098ea69f1d0e20149d 327 326 2023-08-31T05:58:18Z C7X 9 wikitext text/x-wiki Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting.<ref name="RichterAczel74">Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref>^{Section 6} The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals (see section "[[#(+η)-stability|(+η)-stability]]") and more, but not yet \(\Pi^1_1\)-reflecting.<ref>To be proved on this page</ref> In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in [[Ordinal collapsing function|OCFs]] for ordinal analyses of systems beyond [[Kripke-Platek set theory]] with full reflection. This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref name="RichterAczel74" />^{Section 6} ==(+η)-stability== Theorem: Let \(n<\omega\). If \(\alpha\) is \((+n+1)\)-stable, then \(\alpha\) is \(\Pi^1_0\)-reflecting on the class of \((+n)\)-stable ordinals below \(\alpha\). Proof: Assume \(\alpha\) is (+2)-stable and \(\phi(\vec x)\) is a first-order formula with parameters from \(L_\alpha\) such that \(L_\alpha\vDash\phi(\vec x)\). \(L_{\alpha+2}\) satisfies \(\exists\gamma(\phi^{L_\gamma}(\vec x)\land L_\gamma\prec_{\Sigma_1}L_{\gamma+1})\) with \(\alpha\) as a witness of such a \(\gamma\) (is stability well-behaved in \(L_{\gamma+2}\), a successor stage of \(L\)?), and this is a \(\Sigma_1\) formula, so by \(L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\), \(L_\alpha\) satisfies this as well. Then there is a \(\gamma<\alpha\) such that \(\phi^{L_\gamma}(\vec x)\) and \(\gamma\) is \((+1)\)-stable. QED Corollary: \(\Pi^1_1\)-reflecting ordinals \(\alpha\) are \(\alpha^+\)-stable, therefore they are \(\alpha+3\)-stable. So each \(\Pi^1_1\)-reflecting ordinal is \(\Pi^1_0\)-reflecting on the \((+2)\)-stable ordinals below it. 3038817d36ea7ce6de3a684363578dd0055a71ad 0 106 318 286 2023-08-31T00:18:05Z C7X 9 /* Weakly Mahlo */ wikitext text/x-wiki A Mahlo cardinal is a certain type of [[large cardinal]] used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less popular in the literature, weakly Mahlo cardinals are more popular than strongly Mahlo cardinals in apeirological circles, but less popular in the literature. This is essentially due to the fact that weakly Mahlo cardinals are defined in terms of weakly inaccessibles, and strongly Mahlo cardinals are defined in terms of strongly inaccessibles. Essentially, we say a cardinal \( \kappa \) is weakly Mahlo if every club \( C \subseteq \kappa \) contains a regular cardinal. First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{\lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars. Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get [[Gap ordinal|gap ordinals]], and removing it altogether grants you a condition equivalent to regularity! Then, being Mahlo is obtained by adding the condition that such a closure point must be regular: in other words, \( \kappa \) is Mahlo iff. for any function \( f: \kappa \to \kappa \), there is a regular closure point of \( f \) below \( \kappa \). You can see that this is equivalent to Mahloness by noting that the set of closure points of any function is club, and any club is equal to the set of closure points of some function. Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy. == Strongly Mahlo == Strong Mahloness is obtained by replacing "contains a regular cardinal" with "contains a strongly inaccessible cardinal". Clearly any strongly Mahlo cardinal is weakly Mahlo, since every strongly inaccessible cardinal is regular, and the results above can be generalized to show any strongly Mahlo cardinal is strongly hyper-inaccessible. Like the situation between weakly and strongly inaccessible cardinals, \( \mathrm{GCH} \) implies weakly and strongly Mahlo cardinals are the same, while other axioms imply that \( 2^{\aleph_0} \) can be weakly Mahlo. == Ord is Mahlo == "Ord is Mahlo" is an assertion that, as one can likely guess, asserts that every function \( f: \mathrm{Ord} \to \mathrm{Ord} \) has a strongly inaccessible closure point. Clearly, "Ord is Mahlo" implies that there is a proper class of inaccessible cardinals, 1-inaccessible cardinals, and more. However, if \( \kappa \) is Mahlo, then \( V_\kappa \) satisfies "Ord is Mahlo", and thus "Ord is Mahlo" has consistency strength squashed between the inaccessible hierarchy and strongly Mahlo cardinals. Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from [[Reflection principle|here]] that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength. Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to persistently force a statement that isn't true to be true. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo. 89c57e16caa09854f8a15f058e0d3997e7fbb7c6 335 318 2023-08-31T09:54:56Z RhubarbJayde 25 /* Ord is Mahlo */ wikitext text/x-wiki A Mahlo cardinal is a certain type of [[large cardinal]] used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less popular in the literature, weakly Mahlo cardinals are more popular than strongly Mahlo cardinals in apeirological circles, but less popular in the literature. This is essentially due to the fact that weakly Mahlo cardinals are defined in terms of weakly inaccessibles, and strongly Mahlo cardinals are defined in terms of strongly inaccessibles. Essentially, we say a cardinal \( \kappa \) is weakly Mahlo if every club \( C \subseteq \kappa \) contains a regular cardinal. First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{\lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars. Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get [[Gap ordinal|gap ordinals]], and removing it altogether grants you a condition equivalent to regularity! Then, being Mahlo is obtained by adding the condition that such a closure point must be regular: in other words, \( \kappa \) is Mahlo iff. for any function \( f: \kappa \to \kappa \), there is a regular closure point of \( f \) below \( \kappa \). You can see that this is equivalent to Mahloness by noting that the set of closure points of any function is club, and any club is equal to the set of closure points of some function. Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy. == Strongly Mahlo == Strong Mahloness is obtained by replacing "contains a regular cardinal" with "contains a strongly inaccessible cardinal". Clearly any strongly Mahlo cardinal is weakly Mahlo, since every strongly inaccessible cardinal is regular, and the results above can be generalized to show any strongly Mahlo cardinal is strongly hyper-inaccessible. Like the situation between weakly and strongly inaccessible cardinals, \( \mathrm{GCH} \) implies weakly and strongly Mahlo cardinals are the same, while other axioms imply that \( 2^{\aleph_0} \) can be weakly Mahlo. == Ord is Mahlo == "Ord is Mahlo" is an assertion that, as one can likely guess, asserts that every function \( f: \mathrm{Ord} \to \mathrm{Ord} \) has a strongly inaccessible closure point. Clearly, "Ord is Mahlo" implies that there is a proper class of inaccessible cardinals, 1-inaccessible cardinals, and more. However, if \( \kappa \) is Mahlo, then \( V_\kappa \) satisfies "Ord is Mahlo", and thus "Ord is Mahlo" has consistency strength squashed between the inaccessible hierarchy and strongly Mahlo cardinals. Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from [[Reflection principle|here]] that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength. Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to for a statement with real numbers to be false but, in extensions of the universe, possibly necessary. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo. 28de975f3785c9f5c55b5417751b3f2d245fc90d 0 130 320 313 2023-08-31T00:31:27Z C7X 9 "Diagonalizer" terminology not needed, Taranovsky has given this as an example of an ordinal assignment in DoR wikitext text/x-wiki For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An [[ordinal]] \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next [[admissible]] ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy is called non-Gandy or, colloquially, bad. The least bad ordinal is equal to the least ordinal which is \(\Sigma^1_1\)-reflecting<ref>R. Gostanian. The Next Admissible Ordinal. Ann. Math. Logic, 17:171–203, 1979</ref>, which is greater than the least \(\alpha\) which is \(\alpha^+\)-stable, and less than the least \(\alpha\) which is \(\alpha^++1\)-stable.<ref>The Order of Reflection, J. P. Aguilera</ref> The structure of stability at the level of bad ordinals and beyond becomes a lot more complex, due to the highly nonlinear nature of iterated \(\Sigma^1_1\)- and \(\Pi^1_1\)-reflection, and one needing more and more iterations of \(\delta\) to reach the next admissible ordinal. Proper usage of previously ignored, intricate bad ordinal structure is believed to be essential to maximising the strength of ordinal collapsing functions and associated ordinal notations. For example, it is believed that the least bad ordinal is a suitable ordinal to assign to the term \(\Omega\) in Dmytro Taranovsky's "Degrees of Reflection". 4c4ef937f832c4547cd64bbf941dc86edb624ac0 321 320 2023-08-31T00:32:26Z C7X 9 wikitext text/x-wiki For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An [[ordinal]] \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next [[admissible]] ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy is called non-Gandy or, colloquially, bad. The least bad ordinal is equal to the least ordinal which is \(\Sigma^1_1\)-reflecting<ref>R. Gostanian. The Next Admissible Ordinal. Ann. Math. Logic, 17:171–203, 1979</ref>, which is greater than the least \(\alpha\) which is \(\alpha^+\)-stable, and less than the least \(\alpha\) which is \(\alpha^++1\)-stable.<ref>The Order of Reflection, J. P. Aguilera</ref> The structure of stability at the level of bad ordinals and beyond becomes a lot more complex, due to the highly nonlinear nature of iterated \(\Sigma^1_1\)- and \(\Pi^1_1\)-reflection, and one needing more and more iterations of \(\delta\) to reach the next admissible ordinal. Proper usage of previously ignored, intricate bad ordinal structure is believed to be essential to maximising the strength of ordinal collapsing functions and associated ordinal notations. For example, it is believed that the least bad ordinal is a suitable ordinal to assign to the term \(\Omega\) in Dmytro Taranovsky's "Degrees of Reflection".<ref>D. Taranovsky, "Ordinal Notation" (2022). Accessed 30 August 2023.</ref> 31e29a1848f0f5ad400eceb4af55930d78b13012 0 54 322 268 2023-08-31T01:33:25Z EricABQ 5 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. == Extension == This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. 90e6f6d4ae0c37c7b78f5aff819d519cc4ab08b2 324 322 2023-08-31T01:35:43Z EricABQ 5 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. == Extension == This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. == References == 087cbf058d9ced193322caa40773314a49267fbc 0 132 323 2023-08-31T01:35:23Z EricABQ 5 Redirected page to [[Buchholz's psi-functions#Extension]] wikitext text/x-wiki #REDIRECT [[Buchholz's psi-functions#Extension]] 11a511990a03a56b019cc2e61af88acd2d222e85 0 112 325 264 2023-08-31T04:11:16Z C7X 9 Possibly less confusing wording wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \) & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 140be2c5e2425f0f1484e13376034c9db7247a2a 333 325 2023-08-31T06:12:51Z C7X 9 /* List */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 4a7d9176dde4cee2ac1ac9955a32e2276535195e 0 69 328 205 2023-08-31T06:02:38Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997), implicit in section 3. Accessed 29 August 2023.</ref><ref>T. J. Carlson, "Elementary patterns of resemblance", corollary 6.12. Annals of Pure and Applied Logic vol. 108 (2001), pp.19--77.</ref><ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. ac06c415e21e00429a56a54378f4891505d1d402 329 328 2023-08-31T06:03:27Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997), implicit in section 3. Accessed 29 August 2023.</ref><ref name="ElementaryPatterns" />^{corollary 6.12}<ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. d8c69e21d65dd48e71330a47a34b360aad5bf631 330 329 2023-08-31T06:05:39Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref>T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997), implicit in section 3. Accessed 29 August 2023.</ref><ref name="ElementaryPatterns" />^{corollary 6.12}<ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Reflection criterion== Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns.{{citation needed}} ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. 3de50cc9e84dea1de44c59f64a0899b857bcba7a 331 330 2023-08-31T06:06:35Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref name="OrdinalArithmeticSigmaOne">T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997). Accessed 29 August 2023.</ref>^{implicit in section 3}<ref name="ElementaryPatterns" />^{corollary 6.12}<ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Reflection criterion== Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns.(I think <ref name="OrdinalArithmeticSigmaOne" /> is a citation) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. c497b8ee81c52049eff819f4268bcadffbcd9bec 332 331 2023-08-31T06:12:49Z C7X 9 /* Reflection criterion */ wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref name="OrdinalArithmeticSigmaOne">T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997). Accessed 29 August 2023.</ref>^{implicit in section 3}<ref name="ElementaryPatterns" />^{corollary 6.12}<ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Reflection criterion== Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns. (I think <ref name="OrdinalArithmeticSigmaOne" /> is a citation) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. 30b079333f36ccb967811985b04db55d1e78d47e 0 121 334 290 2023-08-31T09:25:27Z RhubarbJayde 25 wikitext text/x-wiki The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that \(N \subseteq V_\alpha\) and, for all \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct. <nowiki>An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\).</nowiki> This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Alternate meaning == An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals. Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. 5aeffb7ffc3be685b754a5b62585998f7914f985 0 115 336 271 2023-08-31T10:00:17Z C7X 9 wikitext text/x-wiki Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail of ordinal analysis, and many believe it can be done using [[Bashicu matrix system|BMS]]. ==Reverse mathematics== One of the primary interests regarding \(Z_2\) is the study of its subsystems, rather than the whole. This is part of a program called reverse mathematics. Since rational numbers, real numbers, complex numbers, continuous functions on the reals, countable groups, and more can be defined in the language of second-order arithmetic, it turns out many classical theorems in number theory, real analysis, topology, abstract algebra and group theory are provable in \(Z_2\), and most even in weak subsystems! The "big five" are the following:<ref>Subsystems of Second Order Arithmetic, Simpson, S.G., ''Perspectives in Logic'', 2009, ''Cambridge University Press''</ref> * \(\mathrm{RCA}_0\): recursive comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^0_1\)-formulae and induction restricted to \(\Sigma^0_1\)-formulae. \(\mathrm{RCA}_0\) has proof-theoretic ordinal [[Omega^omega|\(\omega^\omega\)]], and it can prove the following famous results: the Baire category theorem, the intermediate value theorem, the soundness theorem, the existence of an algebraic closure of a countable field, the existence of a unique real closure of a countable ordered field. * \(\mathrm{WKL}_0\): weak König's lemma, i.e. \(\mathrm{RCA}_0\) with the additional axiom "every infinite binary tree has an infinite branch" \(\mathrm{WKL}_0\) has the same proof-theoretic ordinal as \(\mathrm{RCA}_0\), but is able to prove some non-induction related theorems which \(\mathrm{RCA}_0\) can't, such as: the Heine/Borel covering lemma, every continuous real-valued function on [0, 1], or even any compact metric space, is bounded, the local existence theorem for solutions of (finite systems of) ordinary differential equations, Gödel’s completeness theorem, every countable commutative ring has a prime ideal and Brouwer’s fixed point theorem. * \(\mathrm{ACA}_0\): arithmetical comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^1_0\)-formulae \(\mathrm{ACA}_0\) and Peano arithmetic have the same first-order consequences and thus the same proof-theoretic ordinal: namely, [[Epsilon numbers|\(\varepsilon_0\)]]. Not much has been said regarding \(\mathrm{ACA}_0\)'s ordinary, non-number-theoretical consequences. * \(\mathrm{ATR}_0\): arithmetical transfinite recursion, i.e. \(\mathrm{ACA}_0\) with the additional axiom "every arithmetical operator can be iterated along any countable well-ordering" \(\mathrm{ATR}_0\) has the proof-theoretic ordinal [[Feferman-Schütte ordinal|\(\Gamma_0\)]], which is part of the reason why the ordinal in question is claimed to be the limit of what can be predicatively defined. \(\mathrm{ATR}_0\) can prove the following: any two countable well orderings are comparable, any two countable reduced Abelian p-groups which have the same Ulm invariants are isomorphic, and that every uncountable closed, or analytic, set has a perfect subset. * \(\Pi^1_1 \mathrm{-CA}_0\): \(\Pi^1_1\)-comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Pi^1_1\)-formulae \(\Pi^1_1 \mathrm{-CA}_0\) has a significantly higher proof-theoretic ordinal than the previous entries - namely, [[Buchholz ordinal|\(\psi_0(\Omega_\omega)\)]]. It can prove the following: every countable Abelian group is the direct sum of a divisible group and a reduced group, the Cantor/Bendixson theorem, a set is Borel iff it and its complement are analytic, any two disjoint analytic sets can be separated by a Borel set, coanalytic uniformization, and more. There are also even stronger systems such as \(\Pi^1_1 \mathrm{-TR}_0\), which is \(\Pi^1_1 \mathrm{-CA}_0\) with the axiom "every \(\Pi^1_1\)-definable operator can be iterated along any countable well-ordering", \(\Pi^1_2 \mathrm{-CA}_0\), and more. The former has proof-theoretic ordinal [[Extended Buchholz ordinal|EBO]], while the latter's proof-theoretic ordinal hasn't been precisely calibrated but has been bound.<ref>Determinacy and \(\Pi^1_1\) transfinite recursion along \(\omega\), Takako Nemoto, 2011</ref><ref>An ordinal analysis of \(\Pi_1\)-Collection, Toshiyasu Arai, 2023</ref> 39070f6ffb4b1e4dd114fa67d59d220daa48a0be 0 133 337 2023-08-31T11:05:06Z RhubarbJayde 25 Created page with "The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \i..." wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretation, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies. This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\). If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume \(\kappa\) is measurable, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). \( Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. 45b9d39cf9939c2fad9df7a873bb6cce34e41463 338 337 2023-08-31T11:05:56Z RhubarbJayde 25 /* Alternate characterisations */ wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretation, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies. This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\). If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume \(\kappa\) is measurable, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. 65eb752433249e368cd4183cc677252df81544f5 339 338 2023-08-31T11:06:46Z RhubarbJayde 25 /* Alternate characterisations */ wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretation, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies. This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\). If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. bc10a742efe3280cb970973ab38cd2f5192e82bb 340 339 2023-08-31T11:09:51Z RhubarbJayde 25 /* Alternate characterisations */ wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretation, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies. This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\). If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself. 108fe084c402403bb0c344a34ec8118e6fa84a88 0 134 341 2023-08-31T11:13:16Z RhubarbJayde 25 Redirected page to [[Kripke-Platek set theory]] wikitext text/x-wiki #REDIRECT [[Kripke-Platek set theory]] 1b38a9a73989abdfa33338812817bb79423a75af 0 120 342 281 2023-08-31T11:13:40Z RhubarbJayde 25 wikitext text/x-wiki Kripke-Platek set theory, commonly abbreviated KP, is a weak foundation of set theory used to define admissible ordinals, which are immensely important in ordinal analysis and \(\alpha\)-recursion theory. In terms of proof-theoretic strength, its proof-theoretic ordinal is the [[Bachmann-Howard ordinal|BHO]], and it is thus intermediate between [[Second-order arithmetic|\(\mathrm{ATR}_0\)]] and [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]]. The axioms of KP are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of induction: transfinite induction along the \(\in\)-relation (this implies the axiom of foundation) * Axiom of empty set: There exists a set with no members. * Axiom of pairing: If x, y are sets, then so is {x, y}. * Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x. * Axiom of infinity: there is an inductive set. * Axiom of \(\Delta_0\)-separation: Given any set \(X\) and any \(\Delta_0\)-formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of \(\Delta_0\)-collection: If \(\varphi(x,y)\) is a \(\Delta_0\)-formula so that \(\forall x \exists y \varphi(x,y)\), then for all \(X\), there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). These axioms lead to close connections between KP, \(\alpha\)-recursion theory, and the theory of admissible ordinals. A set \(M\) is admissible if, and only if, it satisfies KP. We say \(\alpha\) is admissible if \(L_\alpha\) is admissible. You can see that this holds if and only if \(\alpha > \omega\) is a limit ordinal and, for every \(\Delta_0(L_\alpha)\)-definable \(f: L_\alpha \to L_\alpha\) and \(x \in L_\alpha\), \(f<nowiki>''</nowiki>x \in L_\alpha\) as well. The least admissible ordinal is the [[Church-Kleene ordinal]]. Note that some authors drop the axiom of infinity and consider \(\omega\) an admissible ordinal too. Note that \(\Delta_0\)-collection actually implies \(\Sigma_1\)-collection. Kripke-Platek set theory may be imagined as being the minimal system of set theory which has infinite sets and allows one to do "computable" and "predicative" definitions over sets. This is because the \(\Sigma_1(L_\alpha)\)-definable functions are considered analogues of computable ones, and, in the context of separation, \(\Delta_0\)-formulae are considered predicative or primitive due to not referencing the totality of the universe. 21821966d26d6d667ba133d24f1de5a90fbdd9b6 351 342 2023-08-31T11:28:24Z RhubarbJayde 25 wikitext text/x-wiki Kripke-Platek set theory, commonly abbreviated KP, is a weak foundation of set theory used to define admissible ordinals, which are immensely important in ordinal analysis and \(\alpha\)-recursion theory. In terms of proof-theoretic strength, its proof-theoretic ordinal is the [[Bachmann-Howard ordinal|BHO]], and it is thus intermediate between [[Second-order arithmetic|\(\mathrm{ATR}_0\)]] and [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]]. The axioms of KP are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of induction: transfinite induction along the \(\in\)-relation (this implies the axiom of foundation) * Axiom of empty set: There exists a set with no members. * Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\). * Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\). * Axiom of infinity: there is an inductive set. * Axiom of \(\Delta_0\)-separation: Given any set \(X\) and any \(\Delta_0\)-formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of \(\Delta_0\)-collection: If \(\varphi(x,y)\) is a \(\Delta_0\)-formula so that \(\forall x \exists y \varphi(x,y)\), then for all \(X\), there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). These axioms lead to close connections between KP, \(\alpha\)-recursion theory, and the theory of admissible ordinals. A set \(M\) is admissible if, and only if, it satisfies KP. We say \(\alpha\) is admissible if \(L_\alpha\) is admissible. You can see that this holds if and only if \(\alpha > \omega\) is a limit ordinal and, for every \(\Delta_0(L_\alpha)\)-definable \(f: L_\alpha \to L_\alpha\) and \(x \in L_\alpha\), \(f<nowiki>''</nowiki>x \in L_\alpha\) as well. The least admissible ordinal is the [[Church-Kleene ordinal]]. Note that some authors drop the axiom of infinity and consider \(\omega\) an admissible ordinal too. Note that \(\Delta_0\)-collection actually implies \(\Sigma_1\)-collection. Kripke-Platek set theory may be imagined as being the minimal system of set theory which has infinite sets and allows one to do "computable" and "predicative" definitions over sets. This is because the \(\Sigma_1(L_\alpha)\)-definable functions are considered analogues of computable ones, and, in the context of separation, \(\Delta_0\)-formulae are considered predicative or primitive due to not referencing the totality of the universe. ea73f6af1a6f3e6a888a3e48037c7ba7d6bebb35 0 135 343 2023-08-31T11:22:53Z RhubarbJayde 25 Redirected page to [[Second-order arithmetic]] wikitext text/x-wiki #REDIRECT [[Second-order arithmetic]] 32bad483c9dfdfde663cd1bf1c136170e814da05 0 136 344 2023-08-31T11:25:31Z RhubarbJayde 25 Created page with "ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as [[Kripke-Platek set theory|KP]] and [[Second-order arithmetic|Z2]]. The axioms are the following: * Axiom of extensionality: two sets are the same if and only if they have the..." wikitext text/x-wiki ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as [[Kripke-Platek set theory|KP]] and [[Second-order arithmetic|Z2]]. The axioms are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of regularity: for all \(x\), if \(x \neq \emptyset\), then there is \(y \in x\) so that \(y \cap x = \emptyset\). * Axiom of separation: Given any set \(X\) and any formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\). * Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\). * Axiom of replacement: For all \(X\), if \(\varphi(x,y)\) is a formula so that \(\forall x \in X \exists! y \varphi(x,y)\), then there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). * Axiom of infinity: there is an inductive set. * Axiom of powerset: Given any set \(x\), \(\{X: X \subseteq x\}\) is also a set. * Axiom of choice: Given any set \(X\), there is a function \(f\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). ZF denotes the theory of ZFC, minus the axiom of choice, which is controversial due to consequences such as the [[Banach-Tarski paradox]]. However, ZF also has its own flaws, such as not being able to prove every set has a well-ordering (which is equivalent to the axiom of choice) and not being able to do cardinal arithmetic or even prove cardinals are comparable. \(\mathrm{ZFC}^-\) or \(\mathrm{ZF}^-\) denote the even weaker theories of ZFC or ZF, respectively, minus the axiom of powerset. These both have the same strength as full [[Second-order arithmetic|Z2]]. The even weaker theory of \(\mathrm{ZFC}^{--}\), where separation is restricted to \(\Delta_0\)-formulae, has the same strength as [[Kripke-Platek set theory|KP]]. Gödel's incompleteness theorems guarantee that there are sentences not provable or disprovable in ZFC, if it is consistent. This incompleteness phenomenon is surprisingly pervasive, and includes sentences such as the [[Constructible hierarchy|axiom of constructibility]] \(V = L\), the continuum hypothesis, the generalized continuum hypothesis, the diamond principle, or the existence of a [[Inaccessible cardinal|weakly inaccessible cardinal]]. 2f8811f4faa5bcfa58691c153ba49bb1c9c50e9b 0 137 345 2023-08-31T11:25:49Z RhubarbJayde 25 Redirected page to [[ZFC]] wikitext text/x-wiki #REDIRECT [[ZFC]] 27081853b11c06a74963924e7fc53c7fd512b2b7 0 138 346 2023-08-31T11:26:07Z RhubarbJayde 25 Redirected page to [[ZFC]] wikitext text/x-wiki #REDIRECT [[ZFC]] 27081853b11c06a74963924e7fc53c7fd512b2b7 0 139 347 2023-08-31T11:26:27Z RhubarbJayde 25 Redirected page to [[ZFC]] wikitext text/x-wiki #REDIRECT [[ZFC]] 27081853b11c06a74963924e7fc53c7fd512b2b7 0 140 348 2023-08-31T11:26:47Z RhubarbJayde 25 Redirected page to [[ZFC]] wikitext text/x-wiki #REDIRECT [[ZFC]] 27081853b11c06a74963924e7fc53c7fd512b2b7 0 141 349 2023-08-31T11:27:04Z RhubarbJayde 25 Redirected page to [[ZFC]] wikitext text/x-wiki #REDIRECT [[ZFC]] 27081853b11c06a74963924e7fc53c7fd512b2b7 0 142 350 2023-08-31T11:27:23Z RhubarbJayde 25 Redirected page to [[ZFC]] wikitext text/x-wiki #REDIRECT [[ZFC]] 27081853b11c06a74963924e7fc53c7fd512b2b7 0 143 352 2023-08-31T12:30:55Z RhubarbJayde 25 Created page with "Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every un..." wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (Jensen's covering theorem fails). * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is determined. * \(\aleph_\omega^V\) is regular in \(\L\). * There is a nontrivial elementary embedding \(j: L \to L\). * There is a proper class of nontrivial elementary embedding \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \(|(\kappa^+)^L| = \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. 600b08229ed14da6bf324c127c350099a90514d0 353 352 2023-08-31T12:31:36Z RhubarbJayde 25 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (Jensen's covering theorem fails). * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\). * There is a proper class of nontrivial elementary embedding \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \(|(\kappa^+)^L| = \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. 0cda7f1c7870d740c82567d212bf3e755e2ae3e0 0 144 354 2023-08-31T12:32:22Z RhubarbJayde 25 Redirected page to [[Cardinal]] wikitext text/x-wiki #REDIRECT [[Cardinal]] 432324f2990dd6428fd42f147d8ec52afa9987f7 0 145 355 2023-08-31T12:32:40Z RhubarbJayde 25 Redirected page to [[Ordinal]] wikitext text/x-wiki #REDIRECT [[Ordinal]] c6b1f78d876dc19592fde69804dd3d22fe63c556 0 57 356 296 2023-08-31T12:34:00Z RhubarbJayde 25 wikitext text/x-wiki A set \(M\) is admissible if \((M,\in)\) is a model of [[Kripke-Platek set theory]]. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\), which is itself equivalent to \(\alpha > \omega\), \(\alpha\) being a limit ordinal and \(L_\alpha\) being closed under preimages of \(\alpha\)-recursively enumerable functions.<ref>Admissible Sets and Structures, Barwise, J., ''Perspectives in Logic'', Cambridge University Press.</ref> The least admissible ordinal is [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]], although some authors omit the axiom of infinity from KP and consider [[Omega|\(\omega\)]] to be admissible. cbd07960a679b393b6c6c6ac52143720cf653095 0 36 357 80 2023-08-31T12:37:41Z RhubarbJayde 25 wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] (a one-to-one correspondence) \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). In terms of von Neumann ordinals, this is equivalent to there being some well-ordering on the set whose order-type is finite. The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]] \(\varnothing\), whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. Likewise, a [[cardinal]] is called '''finite''' when it's the cardinality of a finite set. Once again, finite cardinals can be identified with the natural numbers. == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of two finite ordinals is finite. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of two finite cardinals is finite. == External links == * {{Mathworld|Finite Set|author=Barile, Margherita}} * {{Wikipedia|Finite set}} f810ed371108148df3546dd3939366b85ab9c9af 0 146 358 2023-08-31T12:38:21Z RhubarbJayde 25 Redirected page to [[Ordinal#ordinal arithmetic]] wikitext text/x-wiki #REDIRECT [[Ordinal#ordinal%20arithmetic]] a8b75ec295d05bc9f42c9d19780688e7aa7778ee 359 358 2023-08-31T12:38:35Z RhubarbJayde 25 Changed redirect target from [[Ordinal#ordinal arithmetic]] to [[Ordinal#Ordinal arithmetic]] wikitext text/x-wiki #REDIRECT [[Ordinal#Ordinal%20arithmetic]] c8c1dd8ac2cb078bb39962d27a56442872a4a8be 0 147 360 2023-08-31T12:38:50Z RhubarbJayde 25 Redirected page to [[Ordinal#Ordinal arithmetic]] wikitext text/x-wiki #REDIRECT [[Ordinal#Ordinal%20arithmetic]] c8c1dd8ac2cb078bb39962d27a56442872a4a8be 0 148 361 2023-08-31T12:39:02Z RhubarbJayde 25 Redirected page to [[Ordinal#Ordinal arithmetic]] wikitext text/x-wiki #REDIRECT [[Ordinal#Ordinal%20arithmetic]] c8c1dd8ac2cb078bb39962d27a56442872a4a8be 0 20 362 57 2023-08-31T12:40:41Z RhubarbJayde 25 wikitext text/x-wiki In general mathematics, a '''fixed point''' of a function \(f:X\to X\) is any \(x\in X\) such that \(f(x)=x\). If \(f\) is a [[Normal function|normal]] [[ordinal function]], then the fixed points of \(f\) are precisely the closure points of \(f\): that is, for all \(\alpha\), we have \(f(\alpha) = \alpha\) iff, for all \(\beta < \alpha\), \(f(\beta) < \alpha\). Fixed points are useful in the definition and analysis of apeirological notations such as the [[Veblen hierarchy]] or [[Ordinal collapsing function|OCFs]]. 6fdc208fdb1e701b83d5aff5cc2b843465f74162 0 149 363 2023-08-31T12:42:47Z RhubarbJayde 25 Redirected page to [[Countability]] wikitext text/x-wiki #REDIRECT [[Countability]] 83755c693a20bad68152328fe33a6068aa6e2a3f 0 150 364 2023-08-31T12:44:11Z RhubarbJayde 25 Redirected page to [[Buchholz's psi-functions#Extension]] wikitext text/x-wiki #REDIRECT [[Buchholz's psi-functions#Extension]] 11a511990a03a56b019cc2e61af88acd2d222e85 0 151 365 2023-08-31T12:44:38Z RhubarbJayde 25 Redirected page to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 16 366 275 2023-08-31T12:45:55Z RhubarbJayde 25 wikitext text/x-wiki A normal function is an [[ordinal function]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a [[limit ordinal]]. Veblen's fixed point lemma, which is essential for constructing the [[Veblen hierarchy]], guarantees that, not only does every normal function have a [[fixed point]], but the class of fixed points is unbounded and their enumeration function is also normal. 99b392c08b89b67498a47b607c26d02f16d2f99f 0 152 367 2023-08-31T12:46:17Z RhubarbJayde 25 Redirected page to [[Admissible]] wikitext text/x-wiki #REDIRECT [[Admissible]] 26848641085737fe518bc46ea40a4e7bf4cd7b52 0 153 368 2023-08-31T12:46:55Z RhubarbJayde 25 Redirected page to [[Cardinal]] wikitext text/x-wiki #REDIRECT [[Cardinal]] 432324f2990dd6428fd42f147d8ec52afa9987f7 0 154 369 2023-08-31T12:48:12Z RhubarbJayde 25 Redirected page to [[Ordinal collapsing function]] wikitext text/x-wiki #REDIRECT [[Ordinal collapsing function]] 747f329a67b39510b5a80a72b1d0cb75a18c6ce3 0 63 370 166 2023-08-31T12:48:53Z RhubarbJayde 25 wikitext text/x-wiki '''Pair sequence system''' ('''PSS''') is an [[ordinal notation system]] defined by [[BashicuHyudora]]. It is also a [[sequence system]] with sequences of pairs of natural numbers, and an [[expansion system]] with the base of standard form being \( \{((0,0),(1,1),(2,2),...,(n,n)) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: - The parent of a pair \( x \) in \( S \) is the last pair before it with a smaller first element. The ancestors of \( x \) are recursively defined as the parent of \( x \) and the ancestors of the parent of \( x \). Let \( B_0 \) be the subsequence of \( S \) such that the first pair in \( B_0 \) is the last ancestor of the last pair in \( S \) with the ancestor's second element being strictly smaller than the second element of the last pair in \( S \), and the last pair in \( B_0 \) is the second-to-last pair in \( S \). Then let \( G \) be the subsequence of all pairs in \( S \) before \( B_0 \), and let \( B_i \) be \( B_0 \) but with the first element of each pair increased by \( i \) times the difference between the first element of the last pair in \( S \) and the first element of the first pair in \( B_0 \). Then \( S[n]=G+B_0+B_1+B_2+...+B_n \). The order type of PSS is [[Buchholz ordinal|\( \psi_0(\Omega_\omega) \)]] in [[Buchholz's ordinal collapsing function]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of PSS]</ref> PSS is identical to two-row [[Bashicu matrix system]]. f1b557fa0b1cbc398dbd92d7ae543e010058b088 0 155 371 2023-08-31T12:49:10Z RhubarbJayde 25 Redirected page to [[Buchholz's psi-functions]] wikitext text/x-wiki #REDIRECT [[Buchholz's psi-functions]] d16feb993ff3c3edd2198b78aa1ec52158980509 0 156 372 2023-08-31T13:21:38Z RhubarbJayde 25 Created page with "The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states th..." wikitext text/x-wiki The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the axiom of choice, since it implies that there is no well-ordering of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is believed to be very high. Note that the determinacy of every topological game whose payoff set is closed and/or Borel is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest inner model containing both all ordinals and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. Therefore, \(\mathrm{ZFC} + \mathrm{AD}^{L(\mathbb{R})}\) is actually stronger than \(\mathrm{ZF} + \mathrm{AD}^V\), consistency-wise. 109e6619f156bdf8fe29af2f6e0fa68464a73aa3 373 372 2023-08-31T13:27:42Z RhubarbJayde 25 wikitext text/x-wiki The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the [[axiom of choice]], since it implies that there is no [[Well-ordered set|well-ordering]] of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is very high. Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. Therefore, \(\mathrm{ZFC} + \mathrm{AD}^{L(\mathbb{R})}\) is actually stronger than \(\mathrm{ZF} + \mathrm{AD}^V\), consistency-wise. Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). c7cb5fa05756e6efa3e53487e3d1d8d5c64c4000 0 143 374 353 2023-08-31T13:33:16Z RhubarbJayde 25 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (Jensen's covering theorem fails). * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\). * There is a proper class of nontrivial elementary embedding \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\). * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\). While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. 9b8b7c82220c33f5573ffda751b26c0288efee3f 410 374 2023-08-31T22:52:04Z C7X 9 Sourcing wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (Jensen's covering theorem fails).<ref>Any text about Jensen's covering theorem</ref> * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref> * There is a proper class of nontrivial elementary embedding \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\). (Possible source? <ref>W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref>) * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\). (Possible source? <ref>W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref>) While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. b0f9b6d4fa3267155ba6139a37c360722bad303f 0 157 375 2023-08-31T13:46:50Z RhubarbJayde 25 Created page with "A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] =..." wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^\sword\) and \(0^\q\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^\sword\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\q\) is an inner model for a [[Strong cardinal|strong]] cardinal. 94b9781532bbdd1a3f4782dca5da1b489b547ca2 376 375 2023-08-31T13:47:05Z RhubarbJayde 25 wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^\sword\) and \(0^\p\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^\sword\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\p\) is an inner model for a [[Strong cardinal|strong]] cardinal. 5f59a34a17e3fb8442375c564b676599778c647b 385 376 2023-08-31T14:23:23Z RhubarbJayde 25 wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^\sword\) and \(0^\P\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^\sword\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\P\) is an inner model for a [[Strong cardinal|strong]] cardinal. ad2650f7c0bb7e890e50520f443fe16fb5375a36 386 385 2023-08-31T14:23:55Z RhubarbJayde 25 wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^\sword\) and \(0^\text{\textparagraph}\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^\sword\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\text{\textparagraph}\) is an inner model for a [[Strong cardinal|strong]] cardinal. 8316885324f78d83356b46ed5ac4781cedec2bb8 387 386 2023-08-31T14:24:29Z RhubarbJayde 25 Ignore me miserably failing wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^\sword\) and \(0^\mathparagraph\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^\sword\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\mathparagraph\) is an inner model for a [[Strong cardinal|strong]] cardinal. 4853df061d24d7d322256686db47c1798a8a1777 389 387 2023-08-31T14:32:16Z RhubarbJayde 25 wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^\sword\) and \(0^\text{¶}\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^\sword\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\text{¶}\) is an inner model for a [[Strong cardinal|strong]] cardinal. d92b1ccc04db4b04f26bfb91e0c9d08034b406c3 390 389 2023-08-31T14:33:50Z RhubarbJayde 25 wikitext text/x-wiki A sharp for an [[Inner model theory|inner model]] \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of [[Ordinal|ordinals]] may not be able to be covered by sets in \(N\). For example, the sharp for [[Constructible hierarchy|\(L\)]] is [[Zero sharp|\(0^\sharp\)]]. In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] = L(r)\). For example, it is consistent that \(0^\sharp\) exists but \(0^{\sharp \sharp}\) doesn't, in which case \(L[0^\sharp]\) - the smallest inner model containing \(0^\sharp\) - computes successors of singular cardinals correctly, while \(L\) doesn't. Sharps are important in inner model theory because the nonexistence of certain sharps shows that the inner models constructed possess significant fine structure and satisfy covering, as well as other properties which a "core model" should have. One can also define \(x^\sharp\) when \(x\) isn't a real number. Although the precise definition varies, \(x^\sharp\) is often defined as a sharp for \(L[x]\), the smallest inner model which is amenable for \(x\), or as \(L(x)\), the smallest inner model containing \(x\). When \(x\) is a set of ordinals, the ambiguity disappears. Sharps beyond \(x^\sharp\) include \(0^\dagger\), \(0^{sword}\) and \(0^\text{¶}\). \(0^\dagger\) is a sharp for \(L[U]\), where \(U\) is an ultrafilter witnessing some cardinal's [[Measurable|measurability]]. This is a significantly stronger object than \(x^\sharp\) for sets \(x\), because a measurable cardinal has much more consistency strength than an inaccessible cardinal \(\delta\) such that \(x^\sharp\) exists for all \(x \in H_\delta\), which itself has more consistency strength than the existence of \(r^\sharp\) for all reals \(r\). \(0^{sword}\) is a sharp for an inner model \(N\) accommodating a measurable cardinal with nontrivial [[Mitchell rank]], i.e. so that \((V^\kappa / U)^N\) satisfies "\(\kappa\) is measurable". Lastly, \(0^\text{¶}\) is an inner model for a [[Strong cardinal|strong]] cardinal. d87f5d911dea258c09dc67d5dfa985c2b478fe8e 0 158 377 2023-08-31T13:54:55Z RhubarbJayde 25 Created page with "The empty set is a [[set]] with no elements. Its existence can be proven in [[Kripke-Platek set theory]], even without collection, by applying \(\Delta_0\)-separation with a contradictory formula to an infinite set. The existence of an empty set may seem paradoxical to a beginner to set theory, yet it does not pose any definitional issues and is useful. In particular, the empty set is used in the [[Von Neumann ordinal]] system, in which it encodes the number [[0]]. Also,..." wikitext text/x-wiki The empty set is a [[set]] with no elements. Its existence can be proven in [[Kripke-Platek set theory]], even without collection, by applying \(\Delta_0\)-separation with a contradictory formula to an infinite set. The existence of an empty set may seem paradoxical to a beginner to set theory, yet it does not pose any definitional issues and is useful. In particular, the empty set is used in the [[Von Neumann ordinal]] system, in which it encodes the number [[0]]. Also, the empty set is used in the formal definition of [[gap ordinals]]. The empty set is denoted \(\varnothing\) or \(\emptyset\). When working with ordinals, it may be used interchangeably with \(0\). 2b67a4afb9b7463202fb158f5bf634d5da468893 378 377 2023-08-31T13:55:42Z RhubarbJayde 25 wikitext text/x-wiki The empty set is a [[set]] with no elements. Its existence can be proven in [[Kripke-Platek set theory]], even without collection, by applying \(\Delta_0\)-separation with a contradictory formula to an infinite set. The existence of an empty set may seem paradoxical to a beginner to set theory, yet it does not pose any definitional issues and is useful. In particular, the empty set is used in the [[Von Neumann ordinal]] system, in which it encodes the number [[0]]. Also, the empty set is used in the formal definition of [[gap ordinals]]. The empty set is denoted \(\varnothing\), \(\emptyset\) or \(\{\}\). When working with ordinals, it may be used interchangeably with \(0\). 0a4d549eb7c1b44e97f966c5b3228c7e41d92e6c 0 159 379 2023-08-31T13:58:53Z RhubarbJayde 25 Created page with "1 is the next [[Natural numbers|natural number]] after [[0]]. In the system of [[Von Neumann ordinal|Von Neumann ordinals]] and Zermelo's formalization of the natural numbers, it is represented by the set \(0+1 = \{\{\}\}\), while in the logical formalization of natural numbers it is identified with the proper class of singletons. Also, as a Church numeral, it is identified with the [[lambda calculus]] expression \(\lambda f. \lambda x. f(x)\). 1 is the least Additive..." wikitext text/x-wiki 1 is the next [[Natural numbers|natural number]] after [[0]]. In the system of [[Von Neumann ordinal|Von Neumann ordinals]] and Zermelo's formalization of the natural numbers, it is represented by the set \(0+1 = \{\{\}\}\), while in the logical formalization of natural numbers it is identified with the proper class of singletons. Also, as a Church numeral, it is identified with the [[lambda calculus]] expression \(\lambda f. \lambda x. f(x)\). 1 is the least [[Additive principal ordinals|additive principal ordinal]], being equal to \(\omega^0\), and as such is the least ordinal with a nonempty [[Cantor normal form|CNF]] representation. 48c62f7732251a7db85107ee9c89b89b50fa7052 380 379 2023-08-31T14:00:17Z RhubarbJayde 25 wikitext text/x-wiki 1 is the next [[Natural numbers|natural number]] after [[0]]. In the system of [[Von Neumann ordinal|Von Neumann ordinals]] and Zermelo's formalization of the natural numbers, it is represented by the set \(0+1 = \{\{\}\}\), while in the logical formalization of natural numbers it is identified with the proper class of singletons. Also, as a Church numeral, it is identified with the [[lambda calculus]] expression \(\lambda f. \lambda x. f(x)\). 1 is the least [[Additive principal ordinals|additive principal ordinal]], being equal to \(\omega^0\), and as such is the least ordinal with a nonempty [[Cantor normal form|CNF]] representation. It is equal to the identity in the monoid of [[natural numbers]] under multiplication. b98ce82fb05eebd0d52743d7519b7f6e37aeb915 0 46 381 95 2023-08-31T14:01:03Z RhubarbJayde 25 wikitext text/x-wiki The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. In particular, it is equal to the identity in the monoid of [[natural numbers]] under addition. Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). 20bc5affde1c5c0cdbaaf23697c1aeeaf6bb14f5 0 160 382 2023-08-31T14:04:39Z RhubarbJayde 25 Created page with "A set is one of the basic objects in the mathematical discipline of set theory, upon which most of apeirology is built. Although there's no technical way to define a set, a set is usually considered a collection of objects, and visualized as a bag. For example, the bag can be empty, yielding the [[empty set]]. In some systems of set theory, one has urelements, objects which aren't sets - however, a vast majority of set theory is pure set theory where everything eventuall..." wikitext text/x-wiki A set is one of the basic objects in the mathematical discipline of set theory, upon which most of apeirology is built. Although there's no technical way to define a set, a set is usually considered a collection of objects, and visualized as a bag. For example, the bag can be empty, yielding the [[empty set]]. In some systems of set theory, one has urelements, objects which aren't sets - however, a vast majority of set theory is pure set theory where everything eventually reduces to a set. For example, the [[Von Neumann ordinal]] assignment defines the [[natural numbers]] as nested bags, with zero being empty, and taking successor being adding the number to its own bag - that is, \(0 = \emptyset\) and \(a+1 = a \cup \{a\}\). The discipline of set theory has developed various operations for constructing and comparing sets as well as rules for how sets are behaved, and what properties abstract, infinite sets such as [[Ordinal|ordinals]] have. 0426668f25444fe11e6e4fe1ffda7684ac940ddb 0 161 383 2023-08-31T14:21:17Z RhubarbJayde 25 Created page with "A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity:..." wikitext text/x-wiki A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity: For all \(a\), \(a \leq a\). These properties were intended to idealize and generalize the properties of the ordering on the natural numbers. Note that, under their usual ordering, the natural numbers are well-ordered, but the integers, nonnegative rationals, and any real number interval are not. The [[axiom of choice]] implies, however, that any set can be well-ordered, just the prior examples aren't well-ordered with respect to their natural ordering. As mentioned just now, the axiom of choice implies there is a well-order of the reals, but it is not definable, while the [[axiom of determinacy]] implies that there is no well-order on the reals whatsoever. Well-ordered sets are part of the motivation of [[Ordinal|ordinals]], and are used to define their equivalence class definition. In particular, for any well-ordered set \(X\), there is an ordinal \(\alpha\) and a map \(\pi: X \to \alpha\) so that \(x \leq y\) implies \(\pi(x) \leq \pi(y)\): the unique such ordinal is called its order-type. The order-type of the natural numbers is [[Omega|\(\omega\)]], where the map \(\pi\) is just the identity function, and any ordinal is its own order-type. Also, any [[countable]] set has a countable order-type, and any [[finite]] set has a finite order-type (which is necessarily equal to its [[cardinality]], unlike the case with infinite sets). b81fbe7e790a73d1cf2fb95d207d35c90145b2a8 388 383 2023-08-31T14:31:07Z RhubarbJayde 25 wikitext text/x-wiki A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any nonempty \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity: For all \(a\), \(a \leq a\). These properties were intended to idealize and generalize the properties of the ordering on the natural numbers. Note that, under their usual ordering, the natural numbers are well-ordered, but the integers, nonnegative rationals, and any real number interval are not. The [[axiom of choice]] implies, however, that any set can be well-ordered, just the prior examples aren't well-ordered with respect to their natural ordering. As mentioned just now, the axiom of choice implies there is a well-order of the reals, but it is not definable, while the [[axiom of determinacy]] implies that there is no well-order on the reals whatsoever. Well-ordered sets are part of the motivation of [[Ordinal|ordinals]], and are used to define their equivalence class definition. In particular, for any well-ordered set \(X\), there is an ordinal \(\alpha\) and a map \(\pi: X \to \alpha\) so that \(x \leq y\) implies \(\pi(x) \leq \pi(y)\): the unique such ordinal is called its order-type. The order-type of the natural numbers is [[Omega|\(\omega\)]], where the map \(\pi\) is just the identity function, and any ordinal is its own order-type. Also, any [[countable]] set has a countable order-type, and any [[finite]] set has a finite order-type (which is necessarily equal to its [[cardinality]], unlike the case with infinite sets). You can see that, assuming the axiom of choice, the well-foundedness criterion is equivalent to there not being an infinite descending chain. Assume \(S \subseteq X\), and there is no infinite descending chain. Assume towards contradiction that \(S\) has no minimal element. Use AC to define a choice function \(f\) for \(\{X: X \subseteq S\}\). Let \(s_0 = f(S) \in S\), and then \(s_{n+1} = f(\{x \in S: x \leq s_n \land x \neq s_n\})\). \(\{x \in S: x \leq s_n \land x \neq s_n\}\) is always nonempty, because else \(s_n\) would be a minimal element of \(S\). Then \(s\) forms an infinitely descending chain. Contradiction! For the converse, assume that every nonempty subset has a minimal element, and assume \((s_i)_{i = 0}^\infty\) is an infinite sequence with \(s_i \leq s_j\) for \(j < i\). Let \(S = \{s_i: i \in \mathbb{N}\}\). Therefore \(s\) is eventually constant, and so can't be infinitely ''strictly'' decreasing. fc0618d842c27383216bb711ea525a291147e859 0 51 384 199 2023-08-31T14:21:55Z RhubarbJayde 25 wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of [[Well-ordered set|well-ordered sets]]. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order-type to any well-ordered set. The order-type of a well-ordered set is intuitively its "length". In particular, the order-type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order-type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order-type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) There are helpful visual representations for these, namely with sets of lines. For some basic intuition, [https://www.youtube.com/watch?v=SrU9YDoXE88 see this video]. For example, \(\alpha + \beta\) can be visualized as \(\alpha\), followed by a copy of \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over ZFC: "if \(X\) and \(Y\) are well-ordered sets with order-types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order-type \(\alpha + \beta\)". Also, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order-type: the order-type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. 8b5d4e116a6c485be913877836692b86ce5ddbfb 398 384 2023-08-31T22:16:38Z C7X 9 /* Ordinal arithmetic */ wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of [[Well-ordered set|well-ordered sets]]. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order-type to any well-ordered set. The order-type of a well-ordered set is intuitively its "length". In particular, the order-type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order-type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order-type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) There are helpful visual representations for these, namely with sets of lines. For some basic intuition, [https://www.youtube.com/watch?v=SrU9YDoXE88 see this video]. For example, \(\alpha + \beta\) can be visualized as \(\alpha\), followed by a copy of \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over ZFC: "if \(X\) and \(Y\) are well-ordered sets with order-types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order-type \(\alpha + \beta\)". For ordinal multiplication, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). \(\alpha^\beta\) may be described in terms of functions \(f:\beta\to\alpha\) with finite support.<ref>J. G. Rosenstein, ''Linear Orderings'' (1982). Academic Press, Inc.</ref> == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order-type: the order-type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. 936acd63eb2e0ad4325e13ee9c3cd8eb1f8cdf6e 0 162 391 2023-08-31T14:39:38Z RhubarbJayde 25 Created page with "An [[ordinal function]] is continuous iff it is continuous in the order topology on the [[Ordinal|ordinals]]. If one adds the requirement of being increasing, one obtains the [[Normal function|normal functions]]. However, non-normal continuous functions aren't as studied in the literature and have more complex behaviour. One corollary of the [[Well-ordered set|well-foundedness]] of [[Ordinal|ordinals]] is that there is no continuous decreasing ordinal function which is n..." wikitext text/x-wiki An [[ordinal function]] is continuous iff it is continuous in the order topology on the [[Ordinal|ordinals]]. If one adds the requirement of being increasing, one obtains the [[Normal function|normal functions]]. However, non-normal continuous functions aren't as studied in the literature and have more complex behaviour. One corollary of the [[Well-ordered set|well-foundedness]] of [[Ordinal|ordinals]] is that there is no continuous decreasing ordinal function which is not eventually constant. 45ad1f2525aa50eb5ab54072fab95ad521614281 0 163 392 2023-08-31T14:45:14Z RhubarbJayde 25 Created page with "Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it \(\tav\) and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for Reflection principle|reflecti..." wikitext text/x-wiki Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it \(\tav\) and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for [[Reflection principle|reflection principles]], with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the [[Burali–Forti paradox]]), so large it almost "transcended" itself, and associated it metaphysically with God. Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology. 2933cd47ea915ab0d7a2a140f6af819c41433c37 393 392 2023-08-31T14:45:55Z RhubarbJayde 25 wikitext text/x-wiki Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it with the Hebrew later for Tav and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for [[Reflection principle|reflection principles]], with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the [[Burali–Forti paradox]]), so large it almost "transcended" itself, and associated it metaphysically with God. Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology. d4f1f1fa457990295d57fce9e133967280792817 0 164 394 2023-08-31T14:52:31Z RhubarbJayde 25 Created page with "An [[ordinal]] is called a successor if it is equal to \(\alpha + 1\) for some other \(\alpha\). An ordinal \(\alpha\) is a successor if and only if \(\max \alpha\) exists. The smallest successor ordinal is [[1]], which is also the only successor ordinal to also be [[Additive principal ordinals|additively principal]]. The least ordinal that is not a successor, other than [[0]], is [[Omega|\(\omega\)]]. If \(\beta\) is successor, then \(\alpha+\beta\) is also successor f..." wikitext text/x-wiki An [[ordinal]] is called a successor if it is equal to \(\alpha + 1\) for some other \(\alpha\). An ordinal \(\alpha\) is a successor if and only if \(\max \alpha\) exists. The smallest successor ordinal is [[1]], which is also the only successor ordinal to also be [[Additive principal ordinals|additively principal]]. The least ordinal that is not a successor, other than [[0]], is [[Omega|\(\omega\)]]. If \(\beta\) is successor, then \(\alpha+\beta\) is also successor for all \(\alpha\). However, multiplication and exponentiation do not have this property. 1343d50e2871a63796b7a70b809dbba586aa1ca6 0 165 395 2023-08-31T15:05:12Z RhubarbJayde 25 Created page with "In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper..." wikitext text/x-wiki In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper class, and the [[Burali–Forti paradox]] shows that \(\mathrm{Ord}\). Also, all inner models, such as \(L\), are proper classes. Under the equivalence class definition of [[natural numbers]], [[Ordinal|ordinals]] and [[Cardinal|cardinals]], all numbers (other than [[0]]) are proper classes, which is one of the downsides of this method. [[ZFC]] is a strictly first-order theory, and thus a true treatment of proper classes is not possible within it. Instead, one only uses definable classes such as \(V\), \(\mathrm{Ord}\), and uses \(x \in X\) as a shorthand for \(\varphi(x)\), where \(\varphi\) is a first-order theory. Working within a true second-order treatment of proper, including non-definable, classes is a very powerful tool and can prove, for example, the existence of a proper class of worldly cardinals.<ref>[http://web.archive.org/web/20200916182741/https://philippschlicht.github.io/meetings/files/secondOrderSetTheoryBristol.pdf]</ref> 6e6711095a167ebe290d07242f978aff8a90b332 0 166 396 2023-08-31T15:10:12Z RhubarbJayde 25 Created page with "The axiom of infinity is a common mathematical axiom included in theories such as [[Kripke-Platek set theory]] or [[ZFC]]. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that [[Omega|\(\omega\)]] exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define Ordinal|ordinal..." wikitext text/x-wiki The axiom of infinity is a common mathematical axiom included in theories such as [[Kripke-Platek set theory]] or [[ZFC]]. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that [[Omega|\(\omega\)]] exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define [[Ordinal|ordinals]]. For example, \(V_\omega\), the set of [[Hereditarily finite set|hereditarily finite sets]], is a model of [[ZFC]] minus the axiom of infinity. 13fbd04a7c0c4a6898a37300013ddea6575590ff 0 167 397 2023-08-31T15:13:54Z RhubarbJayde 25 Created page with "A set is hereditarily finite if the smallest transitive set containing it is finite. So, an ordinal is hereditarily finite if and only if it is finite. Any hereditarily finite set is finite, but not every finite set is hereditarily finite, e.g. \(\{\omega\}\). One advantage of using hereditarily finite rather than finite sets is that they form a set, rather than a [[proper class]]. The set in question is \(V_\omega = L_\omega\) in the cumulative/Constructible hierarchy..." wikitext text/x-wiki A set is hereditarily finite if the smallest transitive set containing it is finite. So, an ordinal is hereditarily finite if and only if it is finite. Any hereditarily finite set is finite, but not every finite set is hereditarily finite, e.g. \(\{\omega\}\). One advantage of using hereditarily finite rather than finite sets is that they form a set, rather than a [[proper class]]. The set in question is \(V_\omega = L_\omega\) in the cumulative/[[Constructible hierarchy|constructible]] hierarchies, while there is no \(\alpha\) so that \(V_\alpha\) or \(L_\alpha\) contains all the finite sets. 585addc1fdc988d6781628fe33f98939fab136ed 0 136 399 344 2023-08-31T22:17:37Z C7X 9 Citation for name needed wikitext text/x-wiki ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as [[Kripke-Platek set theory|KP]] and [[Second-order arithmetic|Z2]]. The axioms are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of regularity: for all \(x\), if \(x \neq \emptyset\), then there is \(y \in x\) so that \(y \cap x = \emptyset\). * Axiom of separation: Given any set \(X\) and any formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\). * Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\). * Axiom of replacement: For all \(X\), if \(\varphi(x,y)\) is a formula so that \(\forall x \in X \exists! y \varphi(x,y)\), then there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). * Axiom of infinity: there is an inductive set. * Axiom of powerset: Given any set \(x\), \(\{X: X \subseteq x\}\) is also a set. * Axiom of choice: Given any set \(X\), there is a function \(f\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). ZF denotes the theory of ZFC, minus the axiom of choice, which is controversial due to consequences such as the [[Banach-Tarski paradox]]. However, ZF also has its own flaws, such as not being able to prove every set has a well-ordering (which is equivalent to the axiom of choice) and not being able to do cardinal arithmetic or even prove cardinals are comparable. \(\mathrm{ZFC}^-\) or \(\mathrm{ZF}^-\) denote the even weaker theories of ZFC or ZF, respectively, minus the axiom of powerset. These both have the same strength as full [[Second-order arithmetic|Z2]]. The even weaker theory of \(\mathrm{ZFC}^{--}\){{citation needed}}, where separation is restricted to \(\Delta_0\)-formulae, has the same strength as [[Kripke-Platek set theory|KP]]. Gödel's incompleteness theorems guarantee that there are sentences not provable or disprovable in ZFC, if it is consistent. This incompleteness phenomenon is surprisingly pervasive, and includes sentences such as the [[Constructible hierarchy|axiom of constructibility]] \(V = L\), the continuum hypothesis, the generalized continuum hypothesis, the diamond principle, or the existence of a [[Inaccessible cardinal|weakly inaccessible cardinal]]. 094f74b606ec507257cfd41530d600908e2fb267 400 399 2023-08-31T22:18:39Z C7X 9 May sound like ZF proves Banach-Tarski wikitext text/x-wiki ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as [[Kripke-Platek set theory|KP]] and [[Second-order arithmetic|Z2]]. The axioms are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of regularity: for all \(x\), if \(x \neq \emptyset\), then there is \(y \in x\) so that \(y \cap x = \emptyset\). * Axiom of separation: Given any set \(X\) and any formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\). * Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\). * Axiom of replacement: For all \(X\), if \(\varphi(x,y)\) is a formula so that \(\forall x \in X \exists! y \varphi(x,y)\), then there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). * Axiom of infinity: there is an inductive set. * Axiom of powerset: Given any set \(x\), \(\{X: X \subseteq x\}\) is also a set. * Axiom of choice: Given any set \(X\), there is a function \(f\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). ZF denotes the theory of ZFC, minus the axiom of choice, which is a controversial axiom due to consequences such as the [[Banach-Tarski paradox]]. However, ZF also has its own flaws, such as not being able to prove every set has a well-ordering (which is equivalent to the axiom of choice) and not being able to do cardinal arithmetic or even prove cardinals are comparable. \(\mathrm{ZFC}^-\) or \(\mathrm{ZF}^-\) denote the even weaker theories of ZFC or ZF, respectively, minus the axiom of powerset. These both have the same strength as full [[Second-order arithmetic|Z2]]. The even weaker theory of \(\mathrm{ZFC}^{--}\){{citation needed}}, where separation is restricted to \(\Delta_0\)-formulae, has the same strength as [[Kripke-Platek set theory|KP]]. Gödel's incompleteness theorems guarantee that there are sentences not provable or disprovable in ZFC, if it is consistent. This incompleteness phenomenon is surprisingly pervasive, and includes sentences such as the [[Constructible hierarchy|axiom of constructibility]] \(V = L\), the continuum hypothesis, the generalized continuum hypothesis, the diamond principle, or the existence of a [[Inaccessible cardinal|weakly inaccessible cardinal]]. 46e62cdcddfc5e14cd7326347f15a0afb1e4f367 0 133 401 340 2023-08-31T22:19:40Z C7X 9 How is the reflection principle stated for general cumulative hierarchies? wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretation, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies.{{citation needed}} This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\). If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself. 5d38f1e61ebb73d9f0989537316e719b93e27348 402 401 2023-08-31T22:20:39Z C7X 9 /* Definition */ wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretation, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies.{{citation needed}} This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\).<ref>Most set theory texts</ref> If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself. 43f4e2a7a2f6c157e829d7cf8de0975a38a12469 403 402 2023-08-31T22:20:58Z C7X 9 /* Definition */ wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann definition of ordinal, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, you can note that there \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren't of importance in the context of definable subsets of the natural numbers, since all elements of the natural numbers are definable, but they will be if \(Y\) is uncountable, because no uncountable be pointwise definable, and ensure that there aren't just always \(\aleph_0\) definable subsets of a set. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies.{{citation needed}} This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\).<ref>Most set theory texts</ref> If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself. afd8455766e66cb1460cce653667c47aceed3d1a 404 403 2023-08-31T22:24:07Z C7X 9 /* Definition */ wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann definition of ordinal, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, there are \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) are of importance when \(Y\) is uncountable, to ensure that there are more than \(\aleph_0\) definable subsets of \(Y\), but they do not have any effect when \(Y\supseteq\mathbb N\) is countable, since all elements of the natural numbers are definable. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies.{{citation needed}} This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\).<ref>Most set theory texts</ref> If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself. 1520da27e87a56fcc4bf07557d28ac2f20da4ffc 0 105 405 248 2023-08-31T22:27:29Z C7X 9 /* Strongly inaccessible */ wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \aleph_\omega \) are called limit cardinals because they can't be reached from below via finite iterations of the successor operation: if \( \kappa < \aleph_\omega \), then \( \kappa < \aleph_n \) for some \( n \), thus \( \kappa^{+(m)} < \aleph_{n + m} < \aleph_\omega \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence - thus they are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, cardinals such as \( \aleph_1 \), are unreachable from below via mechanisms such as transfinite recursion, and the limit of any countable sequence of countable ordinals is countable - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega\) acts as a suitable diagonalizer over ordinal arithmetic. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. <s>Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers.</s> Weakly inaccessible cardinals were introduced by Hausdorff to try to work on the continuum hypothesis. Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of accountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" ({{citation needed}}, Jonsson cardinals may be small but they have high consistency strength, and they're considered the higher infinite) You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref> == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. b0629642115074c947026abffb59977f3ef91ac2 406 405 2023-08-31T22:34:51Z C7X 9 More specific example of aleph_1's regularity /* Weakly inaccessible */ wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \(\aleph_\alpha}\) for limit ordinal \(\alpha\) are known as limit cardinals, since applying the cardinal successor operator to a cardinal less than \(\aleph_\alpha\) yields a cardinal also less than \(\aleph_\alpha\). (citation for this being the etymology? {{citation needed}}) However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so no system of normal functions can build up a sequence shorter than \( \aleph_1 \) cofinal in \( \aleph_1 \) - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. <s>Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers.</s> Weakly inaccessible cardinals were introduced by Hausdorff to try to work on the continuum hypothesis. Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of accountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" ({{citation needed}}, Jonsson cardinals may be small but they have high consistency strength, and they're considered the higher infinite) You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref> == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. fa08379023800a78134471e810bd1a48a3ef2f85 407 406 2023-08-31T22:35:15Z C7X 9 /* Weakly inaccessible */ wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). (citation for this being the etymology? {{citation needed}}) However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so no system of normal functions can build up a sequence shorter than \( \aleph_1 \) cofinal in \( \aleph_1 \) - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. <s>Both varieties of inaccessible cardinal were introduced to generalize how unreachable \( \omega \) is from the finite numbers.</s> Weakly inaccessible cardinals were introduced by Hausdorff to try to work on the continuum hypothesis. Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of accountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" ({{citation needed}}, Jonsson cardinals may be small but they have high consistency strength, and they're considered the higher infinite) You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref> == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. 825943985b186aa97dcca677670a8982141edac2 408 407 2023-08-31T22:37:03Z C7X 9 Some history /* Strongly inaccessible */ wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). (citation for this being the etymology? {{citation needed}}) However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so no system of normal functions can build up a sequence shorter than \( \aleph_1 \) cofinal in \( \aleph_1 \) - thus \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). Zermelo referred to the strongly inaccessible cardinals including \( \aleph_0 \) as "Grenzzahlen".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref> However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of uncountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite" ({{citation needed}}, Jonsson cardinals may be small but they have high consistency strength, and they're considered the higher infinite) You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref> == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. a7422fa9755f86f39304b55dea9ff2e62d69ca50 0 168 409 2023-08-31T22:42:04Z C7X 9 Created page with "Cardinals are an extension of the natural numbers that describe the size of a set. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.{{citation needed}} The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.{{citation nedede..." wikitext text/x-wiki Cardinals are an extension of the natural numbers that describe the size of a set. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.{{citation needed}} The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.{{citation nededed}} A large cardinal property is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Maybe "Believing the Axioms II"?</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. a5f08ff47e2f9e5ebea2b5c090a988f409cc5dfa 0 112 411 333 2023-08-31T22:54:42Z C7X 9 /* Quantifier complexity */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref>Possible citation? https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments. == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 12e04bf5e7fbe5fec7ef130ec8b96b0021afb65f 413 411 2023-08-31T22:59:36Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref>Possible citation? https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments. == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal dcf457d2330db446cc190711067aa65d3c66ef67 1 169 412 2023-08-31T22:57:06Z C7X 9 /* "Totally stable" */ new section wikitext text/x-wiki == "Totally stable" == Does "totally stable" in "every uncountable cardinal is totally stable" mean \(L_\kappa\prec L\) for every uncountable cardinal \(\kappa\), or \(L_\kappa\prec_{\Sigma_1}L\)? I think the former is proven in Barwise (I'd have to check), and since using "totally stable" to refer to \(L_\kappa\prec_{\Sigma_1}L\) is pretty much only confined to googology I'm not sure [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:57, 31 August 2023 (UTC) 8f31f27920a4ba8012d42d2281fc9cc1729e5646 0 96 414 300 2023-08-31T23:00:32Z C7X 9 wikitext text/x-wiki The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's [[Buchholz's psi-functions|original set]] of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). Connection to Buchholz hydras c80fdebca947ea58be4aaa744583a9b093008b1e 0 56 415 312 2023-08-31T23:01:50Z C7X 9 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries. However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> 345f3db17a55e85a0088cbd3077ac5207207e9d8 416 415 2023-08-31T23:03:07Z C7X 9 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>GS dimensional Veblen extensions</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> 09572e3800ff7e261853a901237757f02293261d 0 11 417 180 2023-08-31T23:04:21Z C7X 9 wikitext text/x-wiki '''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). Since the function is continuous in the order topology, they are the same as the closure points. Using the [[Veblen hierarchy]], they are enumerated as \(\varphi(1,\alpha)\). The least epsilon number is the limit of "predicative" [[Cantor normal form]], since, as we mentioned, it can't be reached from below via base-\(\omega\) exponentiation. And, in general, \(\varphi(1,\alpha+1)\) is the least ordinal that can't be reached from \(\varphi(1,\alpha)\). By Veblen's fixed point lemma, the enumerating function of the epsilon numbers is normal and thus also has fixed points - these are denoted \(\varphi(2,\alpha)\) or \(\zeta_\alpha\). (Use of the letter \(\zeta\) seems a bit difficult to find, for example sometimes it is called \(\kappa_\alpha\): https://mathoverflow.net/questions/243502) By iterating Cantor normal form and the process of taking (common) fixed points, the [[Veblen hierarchy]] is formed. This induces a natural normal form, called Veblen normal form. Its limit is not \(\zeta_0\), but a much larger ordinal, denoted \(\Gamma_0\). And in general, the ordinals that can't be obtained from below via Veblen normal form are called strongly critical. They are important in ordinal analysis. __NOEDITSECTION__ <!-- Remove the section edit links --> a90b54b9dd8f62d37ec3ab83f46f119675a2e262 0 85 418 221 2023-08-31T23:05:21Z C7X 9 wikitext text/x-wiki The '''large Veblen ordinal''', also called the '''great Veblen number''',<ref>Rathjen, https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, p.10</ref> is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,...,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi(1 @ \alpha) \), where \( 1 @ \alpha \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi(1 @ (1,0)) \), which represents the large Veblen ordinal. This system's limit is the [[Bachmann-Howard ordinal]]. 58d69dbc43bcdea26d7c22ced69d0e995a3a3f5a 419 418 2023-08-31T23:05:43Z C7X 9 wikitext text/x-wiki The '''large Veblen ordinal''', also called the '''great Veblen number''',<ref>Rathjen, https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, p.10</ref> is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,...,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi(1 @ \alpha) \), where \( 1 @ \alpha \) denotes a one followed by \( \alpha \) many zeroes. <!--The @ notation comes from xkcd forums--> This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi(1 @ (1,0)) \), which represents the large Veblen ordinal. This system's limit is the [[Bachmann-Howard ordinal]]. b8305d569d0cb5cef40a1a5592dfde834f5d3cad 420 419 2023-08-31T23:07:36Z C7X 9 wikitext text/x-wiki The '''large Veblen ordinal''', also called the '''great Veblen number''',<ref>Rathjen, https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, p.10</ref> is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,\ldots,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi\begin{pmatrix} 1 \\ \alpha \end{pmatrix} \), where \( \begin{pmatrix} 1 \\ \alpha \end{pmatrix} \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi\begin{pmatrix} 1 \\ (1,0) \end{pmatrix} \) (should there be parentheses in the second row?), which represents the large Veblen ordinal.<ref>GS dimensional Veblen extensions</ref> This system's limit is the [[Bachmann-Howard ordinal]]. 73628f90d67d75639081d8a203120ec13933ef72 1 170 421 2023-08-31T23:15:00Z C7X 9 /* "because that would cause a paradox" */ new section wikitext text/x-wiki == "because that would cause a paradox" == This is a description that has popped up on GS occasionally, but I think it may mislead a bit about "why" classes are proper. If you take the class of infinite cardinals at which GCH fails, by Easton's theorem it's consistent that this is the empty set, and it's also consistent that it can be a proper class. Since it's consistently a set, there must be no paradox/(contradiction with ZFC) that makes it a proper class, yet there are valid models in which ZFC holds and it is a proper class. (This example is given by Asaf Karagila [https://math.stackexchange.com/a/2869598 here]). Additionally, no proper class ever contains another proper class (since over ZFC, proper classes are wrappers for formulae), so the second situation cannot happen in order to show that a class is proper. AFAIK proper classes are exactly the classes that contain elements of unbounded rank (i.e. a class \(C\) is proper iff for any ordinal \(\alpha\), there exists a \(b\in C\) such that \(\textrm{rank}(b)>\alpha\)), but I don't think this can be stated in ZFC. If this holds it may be seen as justification for the common phrase "too large to be a set". [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 23:15, 31 August 2023 (UTC) 98603f5fbf5d916e3ea523cf6aa3d7ca06ef8c37 0 143 422 410 2023-09-01T01:44:05Z C7X 9 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (Jensen's covering theorem fails).<ref>Any text about Jensen's covering theorem</ref> * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref> * There is a proper class of nontrivial elementary embeddings \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\). (Possible source? <ref>W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref>) * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\). (Possible source? <ref>W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref>) While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. 8e3ca55ff5059e7b2cdb8f4c529117a5f324b7ea 0 112 423 413 2023-09-01T01:49:54Z C7X 9 Additional complication /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name"Rathjen94">Possible citation? https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments. In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal ebba6ea8bdf457589b5c4759d5c62c921db6b3d1 424 423 2023-09-01T01:50:29Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name"Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments. In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal e2ce7cf228c7666040bcd62a199b95f0d05ccf00 425 424 2023-09-01T01:50:36Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments. In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 3c97048e542d5c5cc71decd3360b1167206e77fd 429 425 2023-09-01T08:15:31Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023.</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 21cf6e735bcb84f019d32c14d2780975261cd741 430 429 2023-09-01T08:22:21Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 96bba1ede667b7eaa1ffcb35484e1d4eebb7c16b 431 430 2023-09-01T08:42:30Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref><!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 5c07ca9f85c7cc8e49a4f35232c3942a90922f6b 461 431 2023-09-02T01:11:01Z C7X 9 /* Use of nonrecursive countable ordinals */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref>^{Maybe not? Look at things around p.11 more}<!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 6e02ceb8a896a74f66a4f178ad9d4d148352385f 0 163 426 393 2023-09-01T05:32:49Z C7X 9 Connection to reflection principles wikitext text/x-wiki Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it with the Hebrew later for Tav and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for [[Reflection principle|reflection principles]], with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the [[Burali–Forti paradox]]), so large it almost "transcended" itself, and associated it metaphysically with God. Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology. ==As justification for reflection== Later authors have connected Cantor's remark that absolute infinity "can not be conceived" to reflection principles. For example, Maddy states:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic, vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 > Hallet ... traces ''reflection'' to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \( V \) is already true of some [\( V_\alpha \)]. c7b4b39406eb4abd7f7418a8a728d06710fc1326 427 426 2023-09-01T05:35:16Z C7X 9 wikitext text/x-wiki Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it with the Hebrew later for Tav and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for [[Reflection principle|reflection principles]], with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the [[Burali–Forti paradox]]), so large it almost "transcended" itself, and associated it metaphysically with God. Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology. ==As justification for reflection== Later authors have connected Cantor's remark that absolute infinity "can not be conceived" to reflection principles. For example, Maddy states:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic, vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces ''reflection'' to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \( V \) is already true of some [\( V_\alpha \)]. 8403130d80e4112d314530c71c1532cf9678d020 0 70 428 185 2023-09-01T05:49:18Z C7X 9 wikitext text/x-wiki The infinite time Turing machines are a powerful method of computation introduced by Joel David Hamkins and Andy Lewis.<ref>Infinite Time Turing Machines, Joel David Hamkins and Andy Lewis, 1998</ref> They augment the normal notion of a Turing machine (first introduced by Alan Turing in his seminal paper<ref>Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". ''Proceedings of the London Mathematical Society''.</ref>), to a hypothetical machine model which can run for infinitely many steps. It separates the tape into three separate tapes - the input, scratch and output tapes. It's possible to define an analogue of the busy beaver function for ITTMs, denoted \(\Sigma_\infty\), which grows significantly faster than the ordinary busy beaver function, even with an oracle, as well as a halting problem for ITTMs, which has higher Turing degree than \(0^{(\alpha)}\) for all \(\alpha < \gamma\), where \(\gamma\) is the second of the following large ITTM-related ordinals: * \(\lambda\) is the supremum of all ordinals which are the output of an ITTM with empty input. * \(\gamma\) is the supremum of all halting times of an ITTM with empty input. * \(\zeta\) is the supremum of all ordinals which are the eventual output of an ITTM with empty input. * \(\Sigma\) is the supremum of all ordinals which are the accidental output of an ITTM with empty input. ITTMs are quote potent computational models, as they are able to decide whether a given relation on \(\mathbb N\) is a well-order or not.<ref>J. D. Hamkins, A. Lewis, "[https://arxiv.org/abs/math/9808093 Infinite Time Turing Machines]", arXiv 9808093 (2008). Accessed 1 September 2023.</ref>^{theorem 2.2} The ITTM theorem says that \(\lambda = \gamma\), and that: * \(\lambda\) is \(\zeta\)-stable (i.e. \(\zeta\)-\(\Sigma_1\)-stable). * \(\zeta\) is the least ordinal which is \(\rho\)-\(\Sigma_2\)-stable for some \(\rho > \zeta\). * \(\Sigma\) is the least ordinal so that \(\zeta\) is \(\Sigma\)-\(\Sigma_2\)-stable. <nowiki>This further shows the computational potency of ITTMs, since the limit of the order-types of well-orders they can compute is much greater than that of normal TMs, i.e. \(\omega_1^{\mathrm{CK}}\).</nowiki> Infinite time Turing machines can themselves be generalized further to \(\Sigma_n\)-machines, with \(\Sigma_2\)-machines being the same as the original.<ref>Friedman, Sy-David & Welch, P. D. (2011). Hypermachines. Journal of Symbolic Logic</ref> 4185231ab8d8208a426c5b090c421bc243fee8fd 0 168 432 409 2023-09-01T11:05:37Z RhubarbJayde 25 wikitext text/x-wiki Cardinals are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> However, in the context of axiom choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is \(\aleph_0\) - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. d5f30804a83611a7a4743a01ba93b5a0f7945886 0 171 433 2023-09-01T11:06:12Z RhubarbJayde 25 Redirected page to [[Reflection principle#Alternate meaning]] wikitext text/x-wiki #REDIRECT [[Reflection principle#Alternate%20meaning]] 10f01dc635953142e161996140eb2c8378352b97 0 172 434 2023-09-01T11:28:04Z RhubarbJayde 25 Created page with "Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and ot..." wikitext text/x-wiki Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''', is known to be ill-founded. == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> 7af7a4c2fd4006aeadf2bcee8f991d7d2818cf41 435 434 2023-09-01T11:39:41Z RhubarbJayde 25 wikitext text/x-wiki Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''' (Main System with Passthrough), is known to be ill-founded. == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> 7a5664bfcdfade30a2fb3bb4625e0a3e3fd86834 0 173 436 2023-09-01T11:41:08Z RhubarbJayde 25 Created page with "Aleph 0, written \(\aleph_0\), is the [[cardinal]] corresponding to the cardinality of the [[natural numbers]]. As an initial ordinal, it is considered the same as \(\omega\), while it may not be the same while in the absense of the [[axiom of choice]]." wikitext text/x-wiki Aleph 0, written \(\aleph_0\), is the [[cardinal]] corresponding to the cardinality of the [[natural numbers]]. As an initial ordinal, it is considered the same as \(\omega\), while it may not be the same while in the absense of the [[axiom of choice]]. cbdc682e6499c68d65aa7ca8cf616ac94da97722 437 436 2023-09-01T11:41:38Z RhubarbJayde 25 wikitext text/x-wiki Aleph 0, written \(\aleph_0\), is the [[cardinal]] corresponding to the cardinality of the [[natural numbers]]. As an initial (von Neumann) [[ordinal]], it is considered the same as [[Omega|\(\omega\)]], while it may not be the same while in the absense of the [[axiom of choice]]. f9d15c588d737dddc5c19aad17967aaf2b6a6eb0 438 437 2023-09-01T11:41:47Z RhubarbJayde 25 wikitext text/x-wiki Aleph 0, written \(\aleph_0\), is the [[cardinal]] corresponding to the cardinality of the [[natural numbers]]. As an initial (von Neumann) [[ordinal]], it is considered the same as [[Omega|\(\omega\)]], while it may not be the same while in the absence of the [[axiom of choice]]. a4769f75363b7060a2eed7e9d7b94996a2819d49 0 174 439 2023-09-01T12:27:32Z RhubarbJayde 25 Created page with "Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bul..." wikitext text/x-wiki Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> The second-order extension of Peano arithmetic is [[second-order arithmetic]], a significantly more expressive system. One subsystem, \(\mathrm{ACA}_0\) (arithmetical comprehension axiom) is first-order conservative over Peano arithmetic, and is first-order categorical: that is, the first-order parts of any two models of \(\mathrm{ACA}_0\) are isomorphic.<ref>Was Sind und was Sollen Die Zahlen?, Dedekind, R., ''Cambridge Library Collection - Mathematics'', Cambridge University Press</ref> However, Peano arithmetic itself is not categorical, and has many nonstandard models. The set of finite von Neumann [[Ordinal|ordinals]], paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto a \cup \{a\}\) is a model of Peano arithmetic, and one of the most "natural" models of Peano arithmetic. Alternatively, the set of Zermelo ordinals, paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto \{a\}\) is a model of Peano arithmetic. However, the natural functions and relations \(+\), \(\cdot\) and \(<\) in this structure are more complex to describe. Peano arithmetic, minus the axiom schema of induction and plus the axiom \(\forall y (y = 0 \lor \exists x (S(x) = y))\) (which is a theorem of Peano arithmetic but requires induction), is known as Robinson arithmetic, and has proof-theoretic ordinal [[Omega|\(\omega\)]]. As mentioned previously, the axiom schema of induction gives Peano arithmetic a majority of its strength, which is shown by the fact that it has proof-theoretic ordinal [[Epsilon numbers|\(\varepsilon_0\)]], famously shown by Gentzen. Similarly, Robinson arithmetic is unable to show the function \(f_\omega\) in the fast-growing hierarchy is total, while the least rank of the fast-growing hierarchy which outgrows all computable functions provably total in Peano arithmetic is \(f_{\varepsilon_0}\). 79ad5425e1ae702446152908fcfc6c390d646dbc 0 175 440 2023-09-01T12:39:46Z RhubarbJayde 25 Created page with "Lambda calculus is a simple system of computation introduced by Alonzo Church. in which functions, and the operations of abstraction and application, act as primitive operations and objects. [[Natural numbers]] can be encoded in the lambda calculus using a system known as Church numerals. It's been proven that lambda calculus and Turing machines are able to compute the same processes, which led to the independently formulated Church-Turing thesis that all Turing-complete..." wikitext text/x-wiki Lambda calculus is a simple system of computation introduced by Alonzo Church. in which functions, and the operations of abstraction and application, act as primitive operations and objects. [[Natural numbers]] can be encoded in the lambda calculus using a system known as Church numerals. It's been proven that lambda calculus and Turing machines are able to compute the same processes, which led to the independently formulated Church-Turing thesis that all Turing-complete methods of computation are equivalent. John Tromp has invented the system of binary lambda calculus, an extremely compact and efficient encoding of lambda calculus, which he used to define a variant of [[Kleene's O]], and used it to calculate the fundamental sequence of [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]]. 8836d26b70669fe47292ceaa7a76d78cef0d2779 0 161 441 388 2023-09-01T12:43:14Z RhubarbJayde 25 wikitext text/x-wiki A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any nonempty \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity: For all \(a\), \(a \leq a\). These properties were intended to idealize and generalize the properties of the ordering on the natural numbers. Note that, under their usual ordering, the natural numbers are well-ordered, but the integers, nonnegative rationals, and any real number interval are not. The [[axiom of choice]] implies, however, that any set can be well-ordered, just the prior examples aren't well-ordered with respect to their natural ordering. As mentioned just now, the axiom of choice implies there is a well-order of the reals, but it is not definable, while the [[axiom of determinacy]] implies that there is no well-order on the reals whatsoever. Well-ordered sets are part of the motivation of [[Ordinal|ordinals]], and are used to define their equivalence class definition. In particular, for any well-ordered set \(X\), there is an ordinal \(\alpha\) and a map \(\pi: X \to \alpha\) so that \(x \leq y\) implies \(\pi(x) \leq \pi(y)\): the unique such ordinal is called its order-type, and such a function \(\pi\) is called an order-isomorphism. The order-type of the natural numbers is [[Omega|\(\omega\)]], where the map \(\pi\) is just the identity function, and any ordinal is its own order-type. Also, any [[countable]] set has a countable order-type, and any [[finite]] set has a finite order-type (which is necessarily equal to its [[cardinality]], unlike the case with infinite sets). You can see that, assuming the axiom of choice, the well-foundedness criterion is equivalent to there not being an infinite descending chain. Assume \(S \subseteq X\), and there is no infinite descending chain. Assume towards contradiction that \(S\) has no minimal element. Use AC to define a choice function \(f\) for \(\{X: X \subseteq S\}\). Let \(s_0 = f(S) \in S\), and then \(s_{n+1} = f(\{x \in S: x \leq s_n \land x \neq s_n\})\). \(\{x \in S: x \leq s_n \land x \neq s_n\}\) is always nonempty, because else \(s_n\) would be a minimal element of \(S\). Then \(s\) forms an infinitely descending chain. Contradiction! For the converse, assume that every nonempty subset has a minimal element, and assume \((s_i)_{i = 0}^\infty\) is an infinite sequence with \(s_i \leq s_j\) for \(j < i\). Let \(S = \{s_i: i \in \mathbb{N}\}\). Therefore \(s\) is eventually constant, and so can't be infinitely ''strictly'' decreasing. 640897c2045ec1d7f3d70efa842f8bb46cef8029 0 176 442 2023-09-01T12:45:03Z RhubarbJayde 25 Created page with "In set theory, the Cartesian product of two sets, \(X\) and \(Y\), is denoted by \(X \times Y\), and is equal to the set of ordered pairs whose first coordinate is an element of \(X\) and whose second coordinate is an element of \(Y\). Cartesian product is used to give an alternate characterisation of being [[infinite]] - that \(X\) is equinumerous with \(X \times X\). A bijection witnessing this is called a pairing function. However, note that, if \(X\) is an infinite [..." wikitext text/x-wiki In set theory, the Cartesian product of two sets, \(X\) and \(Y\), is denoted by \(X \times Y\), and is equal to the set of ordered pairs whose first coordinate is an element of \(X\) and whose second coordinate is an element of \(Y\). Cartesian product is used to give an alternate characterisation of being [[infinite]] - that \(X\) is equinumerous with \(X \times X\). A bijection witnessing this is called a pairing function. However, note that, if \(X\) is an infinite [[well-ordered set]], \(X\) is never order-isomorphic to \(X \times X\), where the order on \(X \times X\) is lexicographical. Cartesian product is also used to formalize the notions of a relation and a function, originally non-set theoretical concepts, within set theory (without urelements). c1db03ce1582d8618d5b6c1547fcf3c03da54905 447 442 2023-09-01T13:07:37Z RhubarbJayde 25 wikitext text/x-wiki In set theory, the Cartesian product of two sets, \(X\) and \(Y\), is denoted by \(X \times Y\), and is equal to the set of ordered pairs whose first coordinate is an element of \(X\) and whose second coordinate is an element of \(Y\). Cartesian product is used to give an alternate characterisation of being [[infinite]] - that \(X\) is equinumerous with \(X \times X\). A bijection witnessing this is called a pairing function. However, note that, if \(X\) is an infinite [[well-ordered set]], \(X\) is never order-isomorphic to \(X \times X\), where the order on \(X \times X\) is lexicographical. Cartesian product is also used to formalize the notions of a relation and a function, originally non-set theoretical concepts, within set theory (without urelements). It's possible to also define general Cartesian products such as \(X \times Y \times Z\), or even Cartesian products of infinitely many sets. bad30c34f4252ada5c2743700992a7f70b9302ff 0 177 443 2023-09-01T12:52:20Z RhubarbJayde 25 Created page with "The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). The cardinality of the disjoint union of two sets i..." wikitext text/x-wiki The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). The cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. efd2705f88d5ea4070415fdb5bb10ae323436e78 444 443 2023-09-01T12:54:02Z RhubarbJayde 25 wikitext text/x-wiki The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. In particular, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, the cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. 211f17d63d1182916695efb99f21371072ebb1b0 0 136 445 400 2023-09-01T12:57:42Z RhubarbJayde 25 wikitext text/x-wiki ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as [[Kripke-Platek set theory|KP]] and [[Second-order arithmetic|Z2]]. The axioms are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of regularity: for all \(x\), if \(x \neq \emptyset\), then there is \(y \in x\) so that \(y \cap x = \emptyset\). * Axiom of separation: Given any set \(X\) and any formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\). * Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\). * Axiom of replacement: For all \(X\), if \(\varphi(x,y)\) is a formula so that \(\forall x \in X \exists! y \varphi(x,y)\), then there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). * [[Axiom of infinity]]: there is an inductive set. * Axiom of powerset: Given any set \(x\), \(\{X: X \subseteq x\}\) is also a set. * [[Axiom of choice]]: Given any set \(X\), there is a function \(f\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). ZF denotes the theory of ZFC, minus the axiom of choice, which is a controversial axiom due to consequences such as the [[Banach-Tarski paradox]]. However, ZF also has its own flaws, such as not being able to prove every set has a well-ordering (which is equivalent to the axiom of choice) and not being able to do cardinal arithmetic or even prove cardinals are comparable. \(\mathrm{ZFC}^-\) or \(\mathrm{ZF}^-\) denote the even weaker theories of ZFC or ZF, respectively, minus the axiom of powerset. These both have the same strength as full [[Second-order arithmetic|Z2]]. The even weaker theory of \(\mathrm{ZFC}^{--}\){{citation needed}}, where separation is restricted to \(\Delta_0\)-formulae, has the same strength as [[Kripke-Platek set theory|KP]]. Gödel's incompleteness theorems guarantee that there are sentences not provable or disprovable in ZFC, if it is consistent. This incompleteness phenomenon is surprisingly pervasive, and includes sentences such as the [[Constructible hierarchy|axiom of constructibility]] \(V = L\), the continuum hypothesis, the generalized continuum hypothesis, the diamond principle, or the existence of a [[Inaccessible cardinal|weakly inaccessible cardinal]]. 763167a9a51245d04e9682be7ba9cc76b490b2c7 0 120 446 351 2023-09-01T12:58:00Z RhubarbJayde 25 wikitext text/x-wiki Kripke-Platek set theory, commonly abbreviated KP, is a weak foundation of set theory used to define admissible ordinals, which are immensely important in ordinal analysis and \(\alpha\)-recursion theory. In terms of proof-theoretic strength, its proof-theoretic ordinal is the [[Bachmann-Howard ordinal|BHO]], and it is thus intermediate between [[Second-order arithmetic|\(\mathrm{ATR}_0\)]] and [[Second-order arithmetic|\(\Pi^1_1 \mathrm{-CA}_0\)]]. The axioms of KP are the following: * Axiom of extensionality: two sets are the same if and only if they have the same elements. * Axiom of induction: transfinite induction along the \(\in\)-relation (this implies the axiom of foundation) * Axiom of empty set: There exists a set with no members. * Axiom of pairing: If \(x\), \(y\) are sets, then so is \(\{x, y\}\). * Axiom of union: For any set \(x\), there is a set \(y\) such that the elements of \(y\) are precisely the elements of the elements of \(x\). * [[Axiom of infinity]]: there is an inductive set. * Axiom of \(\Delta_0\)-separation: Given any set \(X\) and any \(\Delta_0\)-formula \(\varphi(x)\), \(\{x \in X: \varphi(x)\}\) is also a set. * Axiom of \(\Delta_0\)-collection: If \(\varphi(x,y)\) is a \(\Delta_0\)-formula so that \(\forall x \exists y \varphi(x,y)\), then for all \(X\), there is some \(Y\) so that \(\forall x \in X \exists y \in Y \varphi(x,y)\). These axioms lead to close connections between KP, \(\alpha\)-recursion theory, and the theory of admissible ordinals. A set \(M\) is admissible if, and only if, it satisfies KP. We say \(\alpha\) is admissible if \(L_\alpha\) is admissible. You can see that this holds if and only if \(\alpha > \omega\) is a limit ordinal and, for every \(\Delta_0(L_\alpha)\)-definable \(f: L_\alpha \to L_\alpha\) and \(x \in L_\alpha\), \(f<nowiki>''</nowiki>x \in L_\alpha\) as well. The least admissible ordinal is the [[Church-Kleene ordinal]]. Note that some authors drop the axiom of infinity and consider \(\omega\) an admissible ordinal too. Note that \(\Delta_0\)-collection actually implies \(\Sigma_1\)-collection. Kripke-Platek set theory may be imagined as being the minimal system of set theory which has infinite sets and allows one to do "computable" and "predicative" definitions over sets. This is because the \(\Sigma_1(L_\alpha)\)-definable functions are considered analogues of computable ones, and, in the context of separation, \(\Delta_0\)-formulae are considered predicative or primitive due to not referencing the totality of the universe. 5bb0c221e7720c1486e197053c4d3ef79f46fbd2 0 178 448 2023-09-01T13:10:29Z RhubarbJayde 25 Created page with "The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is..." wikitext text/x-wiki The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is trivial and intuitive. For example, one can see that the axiom of choice is equivalent to the assertion that the Cartesian product of any, not necessarily finite, collection of nonempty sets is nonempty. However, it has some consequences that may be counterintuitive. For example, the axiom of choice implies every set can be well-ordered: that is, for every set \(X\), there is a relation on \(X\) which imbues it with the structure necessary for it to be considered a [[well-ordered set]]. All finite sets, and even countable sets, can be trivially well-ordered, but the countable case is unclear. For example, the axiom of choice is highly nonconstructive and doesn't actually tell somebody what that choice function looks like. Similarly, the axiom of choice tells us there is some well-order on the real numbers, but it is a theorem that there is no well-order on the real numbers. The proof that the axiom of choice implies that every set can be well-ordered is relatively simple. Namely, let \(Y\) be the family of subsets of \(X\). Let \(f\) be a choice function for \(Y\). Then define, via transfinite recursion, the ordinal indexed sequence \(a_\xi\) of elements of \(X\) by \(a_\xi = f(X \setminus \{a_\eta: \eta < \xi\})\). Every element of \(X\) shows up somewhere in this sequence. Therefore, define \(\leq\) by \(a_\xi \leq a_\eta\) iff \(\xi \leq \eta\). This is well-defined, and it is a well-order since the ordinals are well-ordered. Furthermore, the axiom of choice implies the law of excluded middle, which means constructivist mathematicians tend to work in ZF rather than ZFC. Lastly, and most famously, the axiom of choice implies the Banach-Tarski paradox. In particular, using the axiom of choice, it's possible to decompose any ball in 3D space into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. This is counterintuitive, but not truly paradoxical as the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. 23bf979855eae47c82aceeb3af6a3d6022a9fdd6 449 448 2023-09-01T13:13:06Z RhubarbJayde 25 wikitext text/x-wiki The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is trivial and intuitive. For example, one can see that the axiom of choice is equivalent to the assertion that the [[Cartesian product]] of any collection of nonempty sets is nonempty. Note that the assertion that the Cartesian product of finitely many nonempty sets is nonempty is obvious, but it's possible to define Cartesian product of infinitely many sets. Despite these simple characteristics, the axiom of choice is not a theorem of ZF, and it has some consequences that may be counterintuitive. For example, the axiom of choice is highly nonconstructive and doesn't actually tell somebody what that choice function looks like. Similarly, the axiom of choice tells us there is some well-order on the real numbers, but it is a theorem that there is no well-order on the real numbers. In general, the axiom of choice implies every set can be well-ordered: that is, for every set \(X\), there is a relation on \(X\) which imbues it with the structure necessary for it to be considered a [[well-ordered set]]. All finite sets, and even countable sets, can be trivially well-ordered, and in most cases this well-ordering will be definable, but the uncountable case is unclear. The proof that the axiom of choice implies that every set can be well-ordered is relatively simple. Namely, let \(Y\) be the family of subsets of \(X\). Let \(f\) be a choice function for \(Y\). Then define, via transfinite recursion, the [[ordinal]] indexed sequence \(a_\xi\) of elements of \(X\) by \(a_\xi = f(X \setminus \{a_\eta: \eta < \xi\})\). Every element of \(X\) shows up somewhere in this sequence. Therefore, define \(\leq\) by \(a_\xi \leq a_\eta\) iff \(\xi \leq \eta\). This is well-defined, and it is a well-order since the ordinals are well-ordered. Furthermore, the axiom of choice implies the law of excluded middle, which means constructivist mathematicians tend to work in ZF rather than ZFC. Lastly, and most famously, the axiom of choice implies the Banach-Tarski paradox. In particular, using the axiom of choice, it's possible to decompose any ball in 3D space into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. This is counterintuitive, but not truly paradoxical as the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. 89debefe1370c5cab86c9b0a302e7308e8991ccf 0 71 450 307 2023-09-01T13:22:00Z RhubarbJayde 25 wikitext text/x-wiki A weakly compact cardinal is a certain kind of [[large cardinal]]. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indescribable. * \(\kappa\) is \(\kappa\)-unfoldable. * The partition property \(\kappa \to (\kappa)^2_2\) holds. Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem. The existence of a weakly compact cardinal is not provable in ZFC, assuming its existence - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, since "\(\kappa\) is weakly compact" is a \(\Pi^1_2\) property of \(V_\kappa\),<ref>Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:[https://doi.org/10.1007%2F978-3-540-88867-3_2 10.1007/978-3-540-88867-3_2]. ISBN 3-540-00384-3</ref> i.e. a \(\Pi_2\) property of \(V_{\kappa+1}\),<ref>J. D. Hamkins, "[https://jdh.hamkins.org/local-properties-in-set-theory/ Local properties in set theory]" (2014), blog post. Accessed 29 August 2023.</ref> a totally reflecting cardinal, or even a \(\Pi^1_2\)-indescribable cardinal, is larger than the least weakly compact cardinal. Note that, unlike the relation between weakly and strongly inaccessible cardinals, and weakly and strongly Mahlo cardinals, strongly compact cardinals are always significantly greater than weakly compact cardinals, both in terms of consistency strength and size. Also, any weakly compact cardinal is necessarily a strong limit, and there is no known weakening which allows \(2^{\aleph_0}\) to be weakly compact, unlike the case with [[Inaccessible cardinal|weakly inaccessible]] and weakly inaccessible cardinals. ==References== <references /> bbf9f537c7c5df4ad7064b4ff3c523d73b8acb4c 0 179 451 2023-09-01T13:25:34Z RhubarbJayde 25 Created page with "A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]], but not \(\..." wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]], but not \(\Pi^2_2\)-indescribable. This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are compact.<ref>https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large. fa2f90f5b3c0d4e6f431a2cc79f5eba06bd8c620 0 180 452 2023-09-01T13:29:59Z RhubarbJayde 25 Created page with "Inner model theory is the study of the "fine structure theory" and construction of inner models, [[proper class]]-sized models of ZFC which satisfy the existence of large cardinals, covering, the generalized continuum hypothesis, and more. The smallest inner model is [[Constructible hierarchy|\(L\)]], which arguably has the most and the most detailed fine structure, but it is unable to accomodate [[measurable]] cardinals, in the sense that no cardinal, even if it really..." wikitext text/x-wiki Inner model theory is the study of the "fine structure theory" and construction of inner models, [[proper class]]-sized models of ZFC which satisfy the existence of large cardinals, covering, the generalized continuum hypothesis, and more. The smallest inner model is [[Constructible hierarchy|\(L\)]], which arguably has the most and the most detailed fine structure, but it is unable to accomodate [[measurable]] cardinals, in the sense that no cardinal, even if it really is measurable, is measurable in \(L\), and thus one needs to find inner models larger than \(L\). Inner model theory has given rise to the study of [[Sharp|sharps]], and many new [[Large cardinal|large cardinal axioms]] were developed for the purpose of inner model theory. c7a1cbbc6b90425b3dd22c221ad43c8688e1af2c 453 452 2023-09-01T13:30:12Z RhubarbJayde 25 wikitext text/x-wiki Inner model theory is the study of the "fine structure theory" and construction of inner models, [[proper class]]-sized models of ZFC which satisfy the existence of large cardinals, covering, the generalized continuum hypothesis, and more. The smallest inner model is [[Constructible hierarchy|\(L\)]], which arguably has the most and the most detailed fine structure, but it is unable to accommodate [[measurable]] cardinals, in the sense that no cardinal, even if it really is measurable, is measurable in \(L\), and thus one needs to find inner models larger than \(L\). Inner model theory has given rise to the study of [[Sharp|sharps]], and many new [[Large cardinal|large cardinal axioms]] were developed for the purpose of inner model theory. 74599ff9d32af37d289ecdb277b23b68a0d206f4 2 181 454 2023-09-01T13:45:54Z RhubarbJayde 25 Created page with "Relativized nonprojectibility, abbreviated REL-NPR, is a systematic extension of a particular characterisation of nonprojectible ordinals. In general, we say \(\alpha\) is a \(\Gamma\)-cardinal iff, for all \(\gamma < \alpha\), there is no surjection \(\pi: \gamma \to \alpha\) with \(\pi \in \Gamma\). This is motivated by the fact that: #\(\alpha\) is a cardinal iff it is a \(V_{\alpha+1}\)-cardinal. #\(\alpha\) is a gap iff it is an \(L_{\alpha+1}\)-cardinal. #\(\alpha..." wikitext text/x-wiki Relativized nonprojectibility, abbreviated REL-NPR, is a systematic extension of a particular characterisation of nonprojectible ordinals. In general, we say \(\alpha\) is a \(\Gamma\)-cardinal iff, for all \(\gamma < \alpha\), there is no surjection \(\pi: \gamma \to \alpha\) with \(\pi \in \Gamma\). This is motivated by the fact that: #\(\alpha\) is a cardinal iff it is a \(V_{\alpha+1}\)-cardinal. #\(\alpha\) is a gap iff it is an \(L_{\alpha+1}\)-cardinal. #\(\alpha\) is \(\Sigma_2\)-admissible iff it is a \(W_{\alpha+2\}\)-cardinal. #\(\alpha\) is nonprojectible iff it is a \(\Sigma_1(L_\alpha)\)-cardinal. #\(\alpha\) is admissible iff it is a \(W_{\alpha+1}\)-cardinal. Where \(W_{\omega \alpha+n} = \Delta_n(L_\alpha)\). Relativized nonprojectibility will be useful in making a "maximal OCF" where one actually manages to collapse practically all recursive structure '''and''' \(\omega_1\), e.g. in an ordinal analysis of \(\mathrm{ZFC}^-\) augmented by the existence of \(\omega_1\). In general, you can assume the existence of \(0^\dagger\) but the nonexistence of \(0^{\dagger \sharp}\), then you're able to collapse \(\omega_1^L\), \(\omega_1^{L[0^\sharp]}\), \(\omega_1^{L[0^{\sharp \sharp}]}\), and then \(\omega_1\) acts as a diagonalizer of \(\alpha \mapsto \omega_\alpha^{L[U]}\). This is super powerful due to the vast order-type of this structure, and leaves many gap-related notions in the dust. 4595ebc745c8dd16611eb1fb06def227d158363e 455 454 2023-09-01T13:46:18Z RhubarbJayde 25 wikitext text/x-wiki Relativized nonprojectibility, abbreviated REL-NPR, is a systematic extension of a particular characterisation of nonprojectible ordinals. In general, we say \(\alpha\) is a \(\Gamma\)-cardinal iff, for all \(\gamma < \alpha\), there is no surjection \(\pi: \gamma \to \alpha\) with \(\pi \in \Gamma\). This is motivated by the fact that: #\(\alpha\) is a cardinal iff it is a \(V_{\alpha+1}\)-cardinal. #\(\alpha\) is a gap iff it is an \(L_{\alpha+1}\)-cardinal. #\(\alpha\) is \(\Sigma_2\)-admissible iff it is a \(W_{\alpha+2}\)-cardinal. #\(\alpha\) is nonprojectible iff it is a \(\Sigma_1(L_\alpha)\)-cardinal. #\(\alpha\) is admissible iff it is a \(W_{\alpha+1}\)-cardinal. Where \(W_{\omega \alpha+n} = \Delta_n(L_\alpha)\). Relativized nonprojectibility will be useful in making a "maximal OCF" where one actually manages to collapse practically all recursive structure '''and''' \(\omega_1\), e.g. in an ordinal analysis of \(\mathrm{ZFC}^-\) augmented by the existence of \(\omega_1\). In general, you can assume the existence of \(0^\dagger\) but the nonexistence of \(0^{\dagger \sharp}\), then you're able to collapse \(\omega_1^L\), \(\omega_1^{L[0^\sharp]}\), \(\omega_1^{L[0^{\sharp \sharp}]}\), and then \(\omega_1\) acts as a diagonalizer of \(\alpha \mapsto \omega_\alpha^{L[U]}\). This is super powerful due to the vast order-type of this structure, and leaves many gap-related notions in the dust. 143a6b08d7c7733d06f56fe1316a1876328e33d8 0 67 456 177 2023-09-01T13:48:31Z RhubarbJayde 25 wikitext text/x-wiki Countability is a key notion in set theory and apeirology. A set is called countable if it has the same size as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. Georg Cantor, the founder of set theory, proved that the set of integers and the set of rational numbers are both countable, by constructing such maps. More famously, Cantor proved that the real numbers and that the powerset of the natural numbers are both uncountable, by assuming there was a map \( f \) and deriving a contradiction. <nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using ordinal arithmetic. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki> Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the continuum hypothesis. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven. e79e8e8e8900986d50dad62824ab375e24d09b48 0 37 457 83 2023-09-01T13:51:37Z RhubarbJayde 25 wikitext text/x-wiki A [[set]] is said to be '''infinite''' when it is not [[finite]]. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any. Any [[countable]] set is infinite, but there are non-countable infinite sets. An [[ordinal]] is called infinite when it is the [[order type]] of an infinite [[well-ordered set]]. Under the von Neumann representation, this is just equivalent to the ordinal itself being infinite, as a set. The smallest infinite ordinal is [[omega|\(\omega\)]]. Every ordinal larger than it is infinite, and every ordinal smaller than it is finite. Likewise, a [[cardinal]] is called '''infinite''' when it is the [[cardinality]] of an infinite set, and this is also equal to the cardinal itself being infinite under the definition of cardinals as initial ordinal. The smallest infinite cardinal is [[aleph 0|\(\aleph_0\)]]. Every cardinal larger than it is infinite, and every cardinal smaller than it is finite. There are various equivalent ways to phrase the definition of an infinite set. Assuming the [[axiom of choice]], a set \(S\) is infinite if and only if * It is '''Dedekind infinite''', that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in [https://en.wikipedia.org/wiki/Bijection bijection]. * It is in bijection with the [[disjoint union]] \(S\sqcup S\). * It is in bijection with the [[Cartesian product]] \(S\times S\). * \(S\) has a countable subset. Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''. Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of \(\omega\), which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. A model of this theory is provided by the set of [[hereditarily finite set]]s. == Properties == * Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite. * The powerset of a infinite set is infinite. * The set difference of an infinite set and a finite set is infinite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by [[0]], and exponentiation with base 0 or [[1]]. * The [[Cardinal arithmetic|sum]], product, or exponentiation of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above. == Infinity == A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits to infinity], [https://en.wikipedia.org/wiki/Series_(mathematics) series], and [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an object in a structure such as the [https://en.wikipedia.org/wiki/Extended_real_number_line extended real numbers]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number \(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\). == External links == * {{Mathworld|Infinite Set}} * {{Mathworld|Infinity}} * {{Wikipedia|Infinite set}} * {{Wikipedia|Infinity}} 103ea682a8455fdd1a775667dd5c6970b9bb3fdd 10 182 458 2023-09-01T13:53:32Z RhubarbJayde 25 Created page with "<noinclude> <languages/> </noinclude><templatestyles src="Disambiguation/styles.css"/> <div class="disambiguation metadata plainlinks"> <div class="disambiguation-image">[[File:Disambig gray.svg|30px|<translate><!--T:1--> disambiguation</translate>]]</div> <div class="disambiguation-text"><translate><!--T:2--> This is a [[<tvar name=cat>Special:MyLanguage/Category:Disambiguation pages</tvar>|disambiguation page]], which lists pages which may be the intended target.</..." wikitext text/x-wiki <noinclude> <languages/> </noinclude><templatestyles src="Disambiguation/styles.css"/> <div class="disambiguation metadata plainlinks"> <div class="disambiguation-image">[[File:Disambig gray.svg|30px|<translate><!--T:1--> disambiguation</translate>]]</div> <div class="disambiguation-text"><translate><!--T:2--> This is a [[<tvar name=cat>Special:MyLanguage/Category:Disambiguation pages</tvar>|disambiguation page]], which lists pages which may be the intended target.</translate> <translate><!--T:3--> If a [[<tvar name=special>Special:Whatlinkshere/{{PAGENAME}}</tvar>|page link]] referred you here, please consider editing it to point directly to the intended page.</translate></div> </div><includeonly>__DISAMBIG__[[Category:Disambiguation pages{{#translation:}}]]</includeonly><noinclude> {{Documentation|content= {{Uses TemplateStyles|Template:Disambiguation/styles.css}} <translate><!--T:5--> Usage:</translate> <code><nowiki>{{Disambiguation}}</nowiki></code> <translate><!--T:4--> This template adds pages to <tvar name=cat>{{ll|Category:Disambiguation pages}}</tvar>.</translate> }} [[Category:Disambiguation templates{{#translation:}}]] [[Category:Categorizing templates{{#translation:}}|{{PAGENAME}}]] [[Category:Info templates{{#translation:}}|{{PAGENAME}}]] </noinclude> 949eeed3530f867ac22b7e66a56934434cd97ea3 459 458 2023-09-01T13:54:18Z RhubarbJayde 25 wikitext text/x-wiki This is a [[<tvar name=cat>Special:MyLanguage/Category:Disambiguation pages</tvar>|disambiguation page]], which lists pages which may be the intended target.</translate> <translate><!--T:3--> If a [[<tvar name=special>Special:Whatlinkshere/{{PAGENAME}}</tvar>|page link]] referred you here, please consider editing it to point directly to the intended page. 29af1e8b86887f854233507e1a1ee77230b6e09f 460 459 2023-09-01T13:56:19Z RhubarbJayde 25 wikitext text/x-wiki This is a disambiguation page, which lists pages which may be the intended target. If a page link referred you here, please consider editing it to point directly to the intended page. 3e1bae9741abea6ee3c05df7d179e09514fdac1f 3 183 462 2023-09-03T00:37:47Z Alemagno12 27 Created page with "== Can't add external links to user page == When I tried to add a link to the [https://googology.neocities.org Googology Server Archive] in my user page, I got the following error: "This action has been automatically identified as harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: External links on userpages" Ca..." wikitext text/x-wiki == Can't add external links to user page == When I tried to add a link to the [https://googology.neocities.org Googology Server Archive] in my user page, I got the following error: "This action has been automatically identified as harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: External links on userpages" Can you add an exception to this rule, or remove it? (Sorry for not asking on Discord - I have a policy of not using it unless absolutely necessary, and it's better for wiki discussion to happen on the wiki anyways) [User:Alemagno12 ale / nish] 00:37, 3 September 2023 (UTC) 4486c51a0e2ca33123f092d32bbfc0d68c5ffc32 463 462 2023-09-03T00:38:34Z Alemagno12 27 Signature issue wikitext text/x-wiki == Can't add external links to user page == When I tried to add a link to the [https://googology.neocities.org Googology Server Archive] in my user page, I got the following error: "This action has been automatically identified as harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: External links on userpages" Can you add an exception to this rule, or remove it? (Sorry for not asking on Discord - I have a policy of not using it unless absolutely necessary, and it's better for wiki discussion to happen on the wiki anyways) [[User:Alemagno12 ale / nish]] 00:37, 3 September 2023 (UTC) 2cb8da8449404d123ce391ae69f1773e4ca76768 464 463 2023-09-03T00:42:06Z Augigogigi 2 reply to nish wikitext text/x-wiki == Can't add external links to user page == When I tried to add a link to the [https://googology.neocities.org Googology Server Archive] in my user page, I got the following error: "This action has been automatically identified as harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: External links on userpages" Can you add an exception to this rule, or remove it? (Sorry for not asking on Discord - I have a policy of not using it unless absolutely necessary, and it's better for wiki discussion to happen on the wiki anyways) [[User:Alemagno12 ale / nish]] 00:37, 3 September 2023 (UTC) idk how to do that so i made you an admin. should work :+1: --→ [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 00:42, 3 September 2023 (UTC) 77f692e332dfc8d298437d52ad4e588c9082754b 3 183 465 464 2023-09-03T00:46:31Z Alemagno12 27 /* Can't add external links to user page */ wikitext text/x-wiki == Can't add external links to user page == When I tried to add a link to the [https://googology.neocities.org Googology Server Archive] in my user page, I got the following error: "This action has been automatically identified as harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: External links on userpages" Can you add an exception to this rule, or remove it? (Sorry for not asking on Discord - I have a policy of not using it unless absolutely necessary, and it's better for wiki discussion to happen on the wiki anyways) [[User:Alemagno12 ale / nish]] 00:37, 3 September 2023 (UTC) idk how to do that so i made you an admin. should work :+1: --→ [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 00:42, 3 September 2023 (UTC) oh wow, ty!! [[User:Alemagno12|ale / nish]] 065d87d90219e7329605419fe1fe15d8fe55fe56 2 184 466 2023-09-03T00:46:50Z Alemagno12 27 Created page with "Currently working on: * [User:Alemagno12/Six_Trials_of_Formal_Analysis Six Trials of Formal Analysis] - personal project to get experience with ordinal notation bijection proofs (under development) * [https://googology.neocities.org Googology Server Archive] (under development) * NotPi - an ordinal notation that goes beyond BMS (coming soon)" wikitext text/x-wiki Currently working on: * [User:Alemagno12/Six_Trials_of_Formal_Analysis Six Trials of Formal Analysis] - personal project to get experience with ordinal notation bijection proofs (under development) * [https://googology.neocities.org Googology Server Archive] (under development) * NotPi - an ordinal notation that goes beyond BMS (coming soon) 8ddeec4358ae1772dba51c7b71278b19db69a4a8 467 466 2023-09-03T00:47:25Z Alemagno12 27 wikitext text/x-wiki Currently working on: * [[User:Alemagno12/Six_Trials_of_Formal_Analysis|Six Trials of Formal Analysis]] - personal project to get experience with ordinal notation bijection proofs (under development) * [https://googology.neocities.org Googology Server Archive] (under development) * NotPi - an ordinal notation that goes beyond BMS (coming soon) 43929227ebe8a4231954b1fd9c0bba70f2315bc3 6 185 468 2023-09-03T04:14:58Z Alemagno12 27 wikitext text/x-wiki See [[User:Alemagno12/Six_Trials_of_Formal_Analysis] 1bb92f3896eb14cab0066511790402ee90738679 6 186 469 2023-09-03T04:16:35Z Alemagno12 27 wikitext text/x-wiki See [[User:Alemagno12/Six_Trials_of_Formal_Analysis]] 0d1a277fb85bf074317a41da590241719f6d717b 6 187 470 2023-09-03T04:36:29Z Alemagno12 27 wikitext text/x-wiki Claims Nish made in 2018 about the growth rates of HPrSS v1 and v2 06e8cee71edc8bf6d661342c6f7948e20ed51f9e 2 188 471 2023-09-03T04:36:59Z Alemagno12 27 Created page with "I've been a googologist/apeirologist for 9 years now, and have made many analyses ever since. However, all of my analyses have been heuristic guesses and analysis tables; I've since learned how to do formal proofs and I've tried to make formal analyses a few times now, but I've always lost motivation pretty quickly, leaving only a draft or an unfinished proof somewhere in the Googology Discord (see e.g [[File:Unfinished proof 1.jpg||this]] or File:Unfinished proof 2.jp..." wikitext text/x-wiki I've been a googologist/apeirologist for 9 years now, and have made many analyses ever since. However, all of my analyses have been heuristic guesses and analysis tables; I've since learned how to do formal proofs and I've tried to make formal analyses a few times now, but I've always lost motivation pretty quickly, leaving only a draft or an unfinished proof somewhere in the Googology Discord (see e.g [[File:Unfinished proof 1.jpg||this]] or [[File:Unfinished proof 2.jpg||this]]). So I'm proposing myself six analysis challenges to become more familiar with formal analyses, which will hopefully motivate me to do more of them in the future: 1. HPrSS v1 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv1) = HPrSSv3(0,3,5,2,5,7,4,7,9,5), from a [[File:HPrSS claim.jpg||2018 heuristic analysis]]) 2. HPrSS v2 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv2) = HPrSSv3(0,3,5,8,10,4), from a [[File:HPrSS claim.jpg||2018 heuristic analysis]]) 3. [TBA] vs [TBA] 4. [TBA] vs [TBA] 5. Dropping Hydra (x = 1) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra) = BM2.3(0,0,0)(1,1,1)(2,2,0)) 6. Dropping Hydra (full) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra(x+1)) = BM2.3(0,0,0)(1,1,1)(2,2,1)^x(2,2,0)) 8d20e00f61f26ae29b5767419ab500dbdde4ec89 472 471 2023-09-03T04:37:30Z Alemagno12 27 wikitext text/x-wiki I've been a googologist/apeirologist for 9 years now, and have made many analyses ever since. However, all of my analyses have been heuristic guesses and analysis tables; I've since learned how to do formal proofs and I've tried to make formal analyses a few times now, but I've always lost motivation pretty quickly, leaving only a draft or an unfinished proof somewhere in the Googology Discord (see e.g [[File:Unfinished proof 1.jpg||this]] or [[File:Unfinished proof 2.jpg||this]]). So I'm proposing myself six analysis challenges to become more familiar with formal analyses, which will hopefully motivate me to do more of them in the future: # HPrSS v1 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv1) = HPrSSv3(0,3,5,2,5,7,4,7,9,5), from a [[File:HPrSS claim.jpg||2018 heuristic analysis]]) # HPrSS v2 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv2) = HPrSSv3(0,3,5,8,10,4), from a [[File:HPrSS claim.jpg||2018 heuristic analysis]]) # [TBA] vs [TBA] # [TBA] vs [TBA] # Dropping Hydra (x = 1) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra) = BM2.3(0,0,0)(1,1,1)(2,2,0)) # Dropping Hydra (full) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra(x+1)) = BM2.3(0,0,0)(1,1,1)(2,2,1)^x(2,2,0)) 5c3f87c60b314ded8173e0bbfa74870f1ed7424e 473 472 2023-09-03T04:42:37Z Alemagno12 27 wikitext text/x-wiki I've been a googologist/apeirologist for 9 years now, and have made many analyses ever since. However, all of my analyses have been heuristic guesses and analysis tables; I've since learned how to do formal proofs and I've tried to make formal analyses a few times now, but I've always lost motivation pretty quickly, leaving only a draft or an unfinished proof somewhere in the Googology Discord (see e.g [[File:Unfinished proof 1.png||this]] or [[File:Unfinished proof 2.jpg||png]]). So I'm proposing myself six analysis challenges to become more familiar with formal analyses, which will hopefully motivate me to do more of them in the future: # HPrSS v1 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv1) = HPrSSv3(0,3,5,2,5,7,4,7,9,5), from a [[File:HPrSS claim.png||2018 heuristic analysis]]) # HPrSS v2 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv2) = HPrSSv3(0,3,5,8,10,4), from a [[File:HPrSS claim.png||2018 heuristic analysis]]) # [TBA] vs [TBA] # [TBA] vs [TBA] # Dropping Hydra (x = 1) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra) = BM2.3(0,0,0)(1,1,1)(2,2,0)) # Dropping Hydra (full) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra(x+1)) = BM2.3(0,0,0)(1,1,1)(2,2,1)^x(2,2,0)) e65a61c99703c67ef5fb845f02a409b40aa6e441 474 473 2023-09-03T04:43:55Z Alemagno12 27 wikitext text/x-wiki I've been a googologist/apeirologist for 9 years now, and have made many analyses ever since. However, all of my analyses have been heuristic guesses and analysis tables; I've since learned how to do formal proofs and I've tried to make formal analyses a few times now, but I've always lost motivation pretty quickly, leaving only a draft or an unfinished proof somewhere in the Googology Discord (see e.g [[:File:Unfinished proof 1.png||this]] or [[:File:Unfinished proof 2.jpg||png]]). So I'm proposing myself six analysis challenges to become more familiar with formal analyses, which will hopefully motivate me to do more of them in the future: # HPrSS v1 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv1) = HPrSSv3(0,3,5,2,5,7,4,7,9,5), from a [[:File:HPrSS claim.png||2018 heuristic analysis]]) # HPrSS v2 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv2) = HPrSSv3(0,3,5,8,10,4), from a [[:File:HPrSS claim.png||2018 heuristic analysis]]) # [TBA] vs [TBA] # [TBA] vs [TBA] # Dropping Hydra (x = 1) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra) = BM2.3(0,0,0)(1,1,1)(2,2,0)) # Dropping Hydra (full) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra(x+1)) = BM2.3(0,0,0)(1,1,1)(2,2,1)^x(2,2,0)) 667c4ee914c96618ecaa165c0d9e5da1f03b7803 475 474 2023-09-03T04:44:58Z Alemagno12 27 sorry for the edit spam, I haven't edited a wiki page in ages and I keep making mistakes x_x wikitext text/x-wiki I've been a googologist/apeirologist for 9 years now, and have made many analyses ever since. However, all of my analyses have been heuristic guesses and analysis tables; I've since learned how to do formal proofs and I've tried to make formal analyses a few times now, but I've always lost motivation pretty quickly, leaving only a draft or an unfinished proof somewhere in the Googology Discord (see e.g [[:File:Unfinished proof 1.png|this]] or [[:File:Unfinished proof 2.png|this]]). So I'm proposing myself six analysis challenges to become more familiar with formal analyses, which will hopefully motivate me to do more of them in the future: # HPrSS v1 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv1) = HPrSSv3(0,3,5,2,5,7,4,7,9,5), from a [[:File:HPrSS claim.png|2018 heuristic analysis]]) # HPrSS v2 by Yukito vs HPrSS v3 by Yukito (expected correspondence: lim(HPrSSv2) = HPrSSv3(0,3,5,8,10,4), from a [[:File:HPrSS claim.png|2018 heuristic analysis]]) # [TBA] vs [TBA] # [TBA] vs [TBA] # Dropping Hydra (x = 1) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra) = BM2.3(0,0,0)(1,1,1)(2,2,0)) # Dropping Hydra (full) by Hyp cos vs BM2.3 by Bashicu and koteitan (expected correspondence: lim(Dropping Hydra(x+1)) = BM2.3(0,0,0)(1,1,1)(2,2,1)^x(2,2,0)) 9aea9d420cd2eea03b8c80e83f5d40901019d113 0 168 476 432 2023-09-03T15:57:31Z RhubarbJayde 25 wikitext text/x-wiki Cardinals are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> However, in the context of axiom choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is \(\aleph_0\) - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. 3baf2884b85bd968965b77f524c9727141f49940 477 476 2023-09-03T15:58:01Z RhubarbJayde 25 wikitext text/x-wiki Cardinals are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> However, in the context of axiom choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is [[Aleph 0|\(\aleph_0\)]] - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. d90b5e1b845fd790885a5d58ed158a2c0ec3319b 488 477 2023-09-03T17:25:31Z RhubarbJayde 25 wikitext text/x-wiki Cardinals are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> However, in the context of axiom choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is [[Aleph 0|\(\aleph_0\)]] - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. The existence of cardinals such as \(\aleph_\omega\) is guaranteed by the axiom of replacement, and these can not exist without the axiom of replacement, since one isn't in general able to take the suprema of arbitrary [[Set|sets]]. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. 12d9cb1d6e8a415aa1b31129511eb0f1c7f8a9e5 493 488 2023-09-03T17:37:15Z RhubarbJayde 25 wikitext text/x-wiki Cardinals (or cardinal numbers) are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. In general, one may informally describe cardinals as numbers, [[finite]] or [[infinite]], which are meant to describe how many objects there are in a collection. The cardinality of a [[set]] is the (unique) cardinal representing its size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the [[axiom of choice]], since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> Here, the cardinality of a set is just the unique equivalence class which the set belongs to. However, in the context of the axiom of choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). The cardinality of a set is then defined as the minimal ordinal which it bijects with, although it might require some effort to show that the cardinality of a set is always an initial ordinal. Using the [[Ordinal#Von Neumann definition|von Neumann]] interpretation of ordinals, and the initial ordinal interpretation of cardinals, one gets that any [[Natural numbers|natural number]] is a cardinal. In particular, under the initial ordinal definition, a cardinal is a natural number if and only if it is finite. Under the equivalence class definition, a cardinal is a natural number if and only if some (furthermore, any) element of it is finite. Again in the initial ordinal definition, every cardinal is an ordinal yet there are many (infinite) ordinals which are not cardinals. However, under the equivalence class definition, no cardinal is an ordinal. Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is [[Aleph 0|\(\aleph_0\)]] - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. d366dbb4ac9ab84960c3842334c0b5eb5f7b5ed9 0 189 478 2023-09-03T16:26:39Z RhubarbJayde 25 Redirected page to [[Aleph 0]] wikitext text/x-wiki #REDIRECT [[Aleph 0]] c386e0597f428e390dac1a6d8f98476d83ac45ac 0 190 479 2023-09-03T16:35:16Z RhubarbJayde 25 Created page with "Supertasks are a hypothetical mechanism which can be used to simulate [[Infinite time Turing machine|infinite time Turing machines]] and may be related to the divergence or convergence of infinite sums. Furthermore, supertasks can be used to draw [[Matchstick diagram|matchstick diagrams]] for [[infinite]] [[Ordinal|ordinals]]." wikitext text/x-wiki Supertasks are a hypothetical mechanism which can be used to simulate [[Infinite time Turing machine|infinite time Turing machines]] and may be related to the divergence or convergence of infinite sums. Furthermore, supertasks can be used to draw [[Matchstick diagram|matchstick diagrams]] for [[infinite]] [[Ordinal|ordinals]]. b463485e6d1ec9b5f61ca359dcf8afbb905d544f 0 191 480 2023-09-03T16:35:22Z RhubarbJayde 25 Created page with "A bijection between two sets, \(X\) and \(Y\), is a "one-to-one pairing" of their elements. Formally, it is a function \(f: X \to Y\) (which can be encoded as a subset of \(X \times Y\)) so that: * Different elements of \(X\) are sent to different elements of \(Y\). * Every element of \(Y\) has some element of \(X\) which is sent to \(Y\). The first property is known as injectivity, or being 1-1, and can be formally be written as \(f(x) = f(y)\) only if \(x = y\). The..." wikitext text/x-wiki A bijection between two sets, \(X\) and \(Y\), is a "one-to-one pairing" of their elements. Formally, it is a function \(f: X \to Y\) (which can be encoded as a subset of \(X \times Y\)) so that: * Different elements of \(X\) are sent to different elements of \(Y\). * Every element of \(Y\) has some element of \(X\) which is sent to \(Y\). The first property is known as injectivity, or being 1-1, and can be formally be written as \(f(x) = f(y)\) only if \(x = y\). The second property is known as surjectivity, or being onto, and can be formally written as, for all \(y \in Y\), there is \(x \in X\) so that \(f(x) = y\). Bijections are used to define [[Cardinal|cardinals]] and cardinality. a46f0bf8d324f77cbd1bfe0184d107303803ccd7 0 9 481 179 2023-09-03T16:48:23Z RhubarbJayde 25 wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the [[natural numbers]] \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. 481e8d821f2765bd99594aefbd5499d9b5182072 485 481 2023-09-03T17:19:18Z RhubarbJayde 25 /* Properties */ wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is considered by some to be the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. d0350daa9cc46cad000f21dcffebb50133e00535 489 485 2023-09-03T17:31:05Z RhubarbJayde 25 wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. The existence of \(\omega\) is guaranteed by the [[axiom of infinity]]. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is considered by some to be the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. 754bfa6b783ce83ef13394d4095358ea1b8f808f 0 51 482 398 2023-09-03T16:53:33Z RhubarbJayde 25 wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of [[Well-ordered set|well-ordered sets]]. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. This is in contrast to the [[Cardinal|cardinals]], which only describe cardinality, and which are applicable to non-well-ordered sets. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order type to any well-ordered set. The order type of a well-ordered set is intuitively its "length". Alternatively, one may think of the order type of a set \(X\) as the smallest ordinal so that the ordinals below are sufficient to number the elements of \(X\), while preserving the order. For example, the order type of a singleton if 1, since pairing 0 with the single element of that set is a [[bijection]] and preserves order, therefore, the ordinals below 1 are sufficient to order-preservingly number the elements of the singleton. For finite sets, the order type and cardinality are equal. In particular, the order type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) There are helpful visual representations for these, namely with [[Matchstick diagram|matchstick diagrams]]. For example, \(\alpha + \beta\) can be visualized as (a diagram for) \(\alpha\), followed by a copy of (a diagram for) \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines, and they therefore have not only the same [[Cardinal|cardinality]] but the same order type. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over [[ZFC]]: "if \(X\) and \(Y\) are well-ordered sets with order types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order type \(\alpha + \beta\)". For ordinal multiplication, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). \(\alpha^\beta\) may be described in terms of functions \(f:\beta\to\alpha\) with finite support.<ref>J. G. Rosenstein, ''Linear Orderings'' (1982). Academic Press, Inc.</ref> Note that, generally, if one of \(\alpha\) and \(\beta\) is infinite, then \(\alpha + \beta\) will have the same cardinality as \(\max(\alpha, \beta)\), but, as we mentioned, not necessarily the same order type. A similar result holds for ordinal multiplication and addition, and it can be shown by "interlacing" well-ordered sets with the respective order types. An exercise is to formally find a [[bijection]] from \(\max(\alpha, \beta)\) to \(\alpha + \beta\), assuming the former is [[infinite]]. == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order type: the order type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. b8c0f3b70a28912d0e491755ba0e65e7e9109b45 490 482 2023-09-03T17:35:05Z RhubarbJayde 25 wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of [[Well-ordered set|well-ordered sets]]. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. This is in contrast to the [[Cardinal|cardinals]], which only describe cardinality, and which are applicable to non-well-ordered sets. ==Von Neumann definition== In a pure set theory such as ZFC, we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order type to any well-ordered set. The order type of a well-ordered set is intuitively its "length". Alternatively, one may think of the order type of a set \(X\) as the smallest ordinal so that the ordinals below are sufficient to number the elements of \(X\), while preserving the order. For example, the order type of a singleton if 1, since pairing 0 with the single element of that set is a [[bijection]] and preserves order, therefore, the ordinals below 1 are sufficient to order-preservingly number the elements of the singleton. For finite sets, the order type and cardinality are equal. In particular, the order type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) Ordinal arithmetic is well-defined by the axioms of union, pairing and replacement. There are helpful visual representations for these, namely with [[Matchstick diagram|matchstick diagrams]]. For example, \(\alpha + \beta\) can be visualized as (a diagram for) \(\alpha\), followed by a copy of (a diagram for) \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines, and they therefore have not only the same [[Cardinal|cardinality]] but the same order type. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over [[ZFC]]: "if \(X\) and \(Y\) are well-ordered sets with order types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order type \(\alpha + \beta\)". For ordinal multiplication, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). \(\alpha^\beta\) may be described in terms of functions \(f:\beta\to\alpha\) with finite support.<ref>J. G. Rosenstein, ''Linear Orderings'' (1982). Academic Press, Inc.</ref> Note that, generally, if one of \(\alpha\) and \(\beta\) is infinite, then \(\alpha + \beta\) will have the same cardinality as \(\max(\alpha, \beta)\), but, as we mentioned, not necessarily the same order type. A similar result holds for ordinal multiplication and addition, and it can be shown by "interlacing" well-ordered sets with the respective order types. An exercise is to formally find a [[bijection]] from \(\max(\alpha, \beta)\) to \(\alpha + \beta\), assuming the former is [[infinite]]. == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order type: the order type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. 3bed0b96b9c05671f7b93ea441546e48acec4c15 513 490 2023-09-06T18:11:25Z RhubarbJayde 25 wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of [[Well-ordered set|well-ordered sets]]. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. This is in contrast to the [[Cardinal|cardinals]], which only describe cardinality, and which are applicable to non-well-ordered sets. The idea of ordinals as a transfinite extension of the counting numbers was first invented by Georg Cantor in the 19th century. ==Von Neumann definition== In a pure set theory such as [[ZFC]], we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order type to any well-ordered set. One is able to define ordinals without this recursive definition. In particular, in ZFC, the two following statements are equivalent to \(\alpha\) being an ordinal. * \(\alpha\) is a transitive [[set]] and all elements of \(\alpha\) are transitive. * \(\alpha\) is a transitive set and the \(\in\) relation restricted to \(\alpha\) is a strict well-order. Using [[Supertask|supertasks]], it is possible to count up to any given countable ordinal in a finite amount of time. For example, to count to \(\omega\), one takes 30 seconds to count from 0 to 1, 15 seconds to count from 1 to 2, 7.5 seconds to count from 2 to 3, 3.75 seconds to count from 3 to 4, and so on. After a minute, all numbers \(< \omega\) will be exhausted. In general, one can count to \(\omega \cdots n\) in \(n\) seconds. By iterating this process another layer, one can count to \(\omega^2\): one takes 1 minute to count from 0 to \(\omega\), 30 seconds to count from \(\omega\) to \(\omega 2\) (so 15 seconds to count from \(\omega\) to \(\omega + 1\), 7.5 seconds to count from \(\omega + 1\) to \(\omega + 2\), and so on), then 15 seconds to count from \(\omega 2\) to \(\omega 3\), and so on. In general, any countable ordinal can be counted to in a finite amount of time. However, it is impossible to count to [[Countability|\(\omega_1\)]] in any finite amount of time. This idea is closely related to [[Matchstick diagram|matchstick diagrams]] - it is possible to draw a diagram for an arbitrary countable ordinal but not \(\omega_1\). The order type of a well-ordered set is intuitively its "length". Alternatively, one may think of the order type of a set \(X\) as the smallest ordinal so that the ordinals below are sufficient to number the elements of \(X\), while preserving the order. For example, the order type of a singleton if 1, since pairing 0 with the single element of that set is a [[bijection]] and preserves order, therefore, the ordinals below 1 are sufficient to order-preservingly number the elements of the singleton. For finite sets, the order type and cardinality are equal. In particular, the order type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) Ordinal arithmetic is well-defined by the axioms of union, pairing and replacement. There are helpful visual representations for these, namely with [[Matchstick diagram|matchstick diagrams]]. For example, \(\alpha + \beta\) can be visualized as (a diagram for) \(\alpha\), followed by a copy of (a diagram for) \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines, and they therefore have not only the same [[Cardinal|cardinality]] but the same order type. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over [[ZFC]]: "if \(X\) and \(Y\) are well-ordered sets with order types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order type \(\alpha + \beta\)". For ordinal multiplication, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). \(\alpha^\beta\) may be described in terms of functions \(f:\beta\to\alpha\) with finite support.<ref>J. G. Rosenstein, ''Linear Orderings'' (1982). Academic Press, Inc.</ref> Note that, generally, if one of \(\alpha\) and \(\beta\) is infinite, then \(\alpha + \beta\) will have the same cardinality as \(\max(\alpha, \beta)\), but, as we mentioned, not necessarily the same order type. A similar result holds for ordinal multiplication and addition, and it can be shown by "interlacing" well-ordered sets with the respective order types. An exercise is to formally find a [[bijection]] from \(\max(\alpha, \beta)\) to \(\alpha + \beta\), assuming the former is [[infinite]]. == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order type: the order type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. f02360f404e33c2f7dc22a5e9960437edf3b0b3d 0 173 483 438 2023-09-03T16:53:49Z RhubarbJayde 25 wikitext text/x-wiki Aleph 0, written \(\aleph_0\) and said aleph null, is the [[cardinal]] corresponding to the cardinality of the [[natural numbers]]. As an initial (von Neumann) [[ordinal]], it is considered the same as [[Omega|\(\omega\)]], while it may not be the same while in the absence of the [[axiom of choice]]. In the context of AC, \(\aleph_0\) is the smallest infinite cardinal and, as such, larger than all numbers considered in googology. It can not be reached from below by any form of arithmetic, and as such may be considered analogous to an [[Inaccessible cardinal|inaccessible]]. Aleph 0 is also the cardinality of the integers, and of any infinite subset of the naturals or integers. Furthermore, Cantor proved, via diagonalization, that, surprisingly, aleph 0 is also the cardinality of the rational numbers. 329ea5b88771d92206b632da8b997c19ba3c4b3e 0 192 484 2023-09-03T17:17:43Z RhubarbJayde 25 Created page with "The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. [[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The questi..." wikitext text/x-wiki The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. [[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The question of whether \(\omega_1\), the least uncountable [[cardinal]], and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the continuum hypothesis. Surprisingly, assuming its consistency, this is neither provable nor disprovable in [[ZFC]]! The existence of an arbitrary set's powerset is not provable from [[Kripke-Platek set theory|KP]], even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC. 38571baec1b555f97b74adb57584bf5156614ff0 497 484 2023-09-03T17:52:28Z RhubarbJayde 25 wikitext text/x-wiki The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. [[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The question of whether \(\omega_1\), the least uncountable [[cardinal]], and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the [[continuum hypothesis]]. Surprisingly, assuming its consistency, this is neither provable nor disprovable in [[ZFC]]! The existence of an arbitrary set's powerset is not provable from [[Kripke-Platek set theory|KP]], even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC. eb859b56888463de272b40b3d1c25b466785009f 0 114 486 270 2023-09-03T17:19:41Z RhubarbJayde 25 wikitext text/x-wiki An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additively principal ordinal is 1 since \(0 + 0 = 1\), and all additively principal ordinals other than 1 are limit ordinals. In particular, you can see from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation) that the additively principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some \(\gamma\). As such, the second infinite additively principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additively principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of those is \(\omega^{\omega^2}\). <nowiki>Additively principal ordinals can be generalized to multiplicatively principal ordinals and exponentially principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicatively principal ordinals are to additively principal ordinals as additively principal ordinals are to limit ordinals. However, exponentially principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the </nowiki>[[epsilon numbers]]. f7cfb11f35d671c59c893988217797b7faea0061 0 193 487 2023-09-03T17:24:14Z RhubarbJayde 25 Created page with "Cantor's diagonal argument is a method for showing the [[Countability|uncountability]] of the set of real numbers. It is a proof by contradiction - one assumes that, towards contradiction, there is a [[bijection]] from the [[natural numbers]] to the real numbers, and then one constructs a real number not in the range of this function, which contradicts surjectivity. It may be rephrased as the assertion that every function from the naturals to the reals is non-surjective,..." wikitext text/x-wiki Cantor's diagonal argument is a method for showing the [[Countability|uncountability]] of the set of real numbers. It is a proof by contradiction - one assumes that, towards contradiction, there is a [[bijection]] from the [[natural numbers]] to the real numbers, and then one constructs a real number not in the range of this function, which contradicts surjectivity. It may be rephrased as the assertion that every function from the naturals to the reals is non-surjective, and in that sense it could be considered a direct proof instead. First, for a real number \(r\), let \(r[n]\) be the \(n\)th digit after the decimal place, in the "lexicographically maximal" decimal expansion of \(r\). For example, \(\pi[0] = 1\) and \(\pi[1] = 4\). Then the proof goes like so: assume \(f: \mathbb{N} \to \mathbb{R}\) is a supposed function enumerating the entirety of the real numbers. Let \(r\) be the real number with whole part 0. Then \(r[n] = 0\) if \(f(n)[n] \neq 0\), and else \(r[n] = 1\). Now, since \(f\) is surjective, there is some \(m\) so that \(f(m) = r\). Then, if \(f(m)[m] \neq 0\), we have \(r[m] = 0\), and if \(f(m)[m] = 0\), we have \(r[m] = 1\). That is, \(r\) and \(f(m)\) disagree on the \(m\)th decimal place after the decimal point, and therefore they can not be the same real number. The proof is known as the "diagonal argument" because of a common visual representation. Namely, one writes out the decimal expansions of all the supposed real numbers on a grid, so that \(f(n)[m]\) is equal to the entry at the \(m+1\)st column and \(n+1\)st row, and then constructs a new real number by "inverting" the decimal expansion of the diagonal. The conclusion can be strengthened to show that just the real interval \([0,1]\) is uncountable. This is because the whole == P(N) == A similar argument can be used to show the uncountability of the [[powerset]] of the natural numbers, \(\mathcal{P}(\mathbb{N})\). Note that, by using binary expansions and characteristic functions, one can see that \([0,1]\) has the same size as \(\mathcal{P}(\mathbb{N})\), with \(0\) corresponding to [[Empty set|\(\emptyset\)]] and \(1\) corresponding to \(\mathbb{N}\). However, the application of the method of diagonalization to \(\mathcal{P}(\mathbb{N})\) has some merit in its own right. Assume \(f: \mathbb{N} \to \mathcal{P}(\mathbb{N})\) is a supposed function enumerating the entirety of the powerset of the natural numbers. We define a set \(X\) like so: \(n \in X\) iff \(n \notin f(n)\). Now, since \(f\) is surjective, there is some \(m\) so that \(f(m) = X\). Then, we have \(m \in X\) iff \(m \notin f(m)\) iff \(m \notin X\). This is impossible! And this can be generalize to show that, for every set \(X\), we have \(2^{|X|} > |X|\). The general form of diagonalization-type arguments is similar to the barber paradox, or, more precisely, Russel's paradox. 85e187323ca49fadad071a6457d9f596938afeec 0 194 491 2023-09-03T17:35:45Z RhubarbJayde 25 Redirected page to [[Aleph 0]] wikitext text/x-wiki #REDIRECT [[Aleph 0]] c386e0597f428e390dac1a6d8f98476d83ac45ac 0 195 492 2023-09-03T17:36:46Z RhubarbJayde 25 Redirected page to [[Epsilon numbers]] wikitext text/x-wiki #REDIRECT [[Epsilon numbers]] 4603d418f29680a0e3b925e67b4471f50e268bb4 0 67 494 456 2023-09-03T17:38:28Z RhubarbJayde 25 wikitext text/x-wiki Countability is a key notion in set theory and apeirology. A set is called countable if it has the same size as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. Georg Cantor, the founder of set theory, proved that the set of integers and the set of rational numbers are both countable, by constructing such maps. More famously, Cantor proved that the real numbers and that the powerset of the natural numbers are both uncountable, by assuming there was a map \( f \) and deriving a contradiction. <nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki> Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the continuum hypothesis. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven. 486458db84fbcba6905de9039bdc3d639af88107 498 494 2023-09-03T17:52:50Z RhubarbJayde 25 wikitext text/x-wiki Countability is a key notion in set theory and apeirology. A set is called countable if it has the same size as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. Georg Cantor, the founder of set theory, proved that the set of integers and the set of rational numbers are both countable, by constructing such maps. More famously, Cantor proved that the real numbers and that the powerset of the natural numbers are both uncountable, by assuming there was a map \( f \) and deriving a contradiction. <nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki> Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the [[continuum hypothesis]]. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven. 235876e36a98cd5e1d5aa35d62d5b267aa53af68 502 498 2023-09-04T11:14:12Z RhubarbJayde 25 wikitext text/x-wiki Countability is a key notion in set theory and apeirology. A set is called countable if it is possible to arrange its elements in a way so that they can be counted off one-by-one. In other words, it has the same [[Cardinal|size]] as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. In other words, there is a [[bijection]] from \( x \) to the natural numbers. However, this may not imply they have the same order-type, if the set is [[Well-ordered set|well-ordered]]. Georg Cantor, the founder of set theory, proved that the set of integers and the set of rational numbers are both countable, by constructing such maps. More famously, Cantor [[Cantor's diagonal argument|proved]] that the real numbers and that the powerset of the natural numbers are both uncountable, by assuming there was a map \( f \) and deriving a contradiction. In terms of ordinals, it is clear that [[Omega|\( \omega \)]]<nowiki> is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki> Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the [[continuum hypothesis]]. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven. 1ffeae49cd6f0b097139920ea2bde571465231f8 0 196 495 2023-09-03T17:46:23Z RhubarbJayde 25 Created page with "An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in [[ZFC]]. Aleph fixed points are large in that they are unreachable from below via the..." wikitext text/x-wiki An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in [[ZFC]]. Aleph fixed points are large in that they are unreachable from below via the aleph operator. However, it is possible that the number of real numbers is an aleph fixed point, or more. Furthermore, the least aleph fixed point has cofinality \(\omega\), which follows from the [[Normal function|normality]] of \(f(\alpha) = \aleph_\alpha\). A regular aleph fixed point is precisely a [[Inaccessible cardinal|weakly inaccessible cardinal]], and, therefore, the [[large cardinal]] hierarchy is beyond the notion of aleph fixed points, the fixed points of their enumeration, and so on, since those can all be proven to exist and are less than the least weakly inaccessible cardinal, if it exists. In most if not all [[Ordinal collapsing function|OCFs]], the collapse of the least aleph fixed point is the [[Extended Buchholz ordinal]], which is why it is sometimes alternately referred to as the OFP, although this is technically a misnomer. fc7258496dc43d424971724e5fe17fdb1257bcff 0 105 496 408 2023-09-03T17:51:10Z RhubarbJayde 25 wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals<ref>Jech, Thomas (2003), ''Set Theory'', Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag</ref>, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called singular. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so \( \aleph_1 \) is not singular - aka regular. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). Zermelo referred to the strongly inaccessible cardinals including \( \aleph_0 \) as "Grenzzahlen".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref> However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of uncountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite". You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref> == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. 79b9d4f2483aedfd5405c48ddee54ee1d3456465 0 197 499 2023-09-03T18:06:05Z RhubarbJayde 25 Created page with "The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the [[axiom of choice]], and \(\aleph_1\) is the smallest [[Countability|uncountable]] cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the con..." wikitext text/x-wiki The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the [[axiom of choice]], and \(\aleph_1\) is the smallest [[Countability|uncountable]] cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the context of the [[axiom of determinacy]], it holds that, for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size, and yet \(2^{\aleph_0} \neq \aleph_1\) (in particular, the two are incomparable, since one needs choice to prove cardinals are linearly ordered). The continuum hypothesis was originally posed by Georg Cantor after his proof of [[Cantor's diagonal argument|the diagonal argument]]. Cantor was unable to prove or disprove the continuum hypothesis, primarily because he was unable to find a [[Well-ordered set|well-order]] of the reals. It is now known that there is no definable well-order on the reals: therefore, this approach can't work. Solving the continuum hypothesis was the first on Hilbert's famous list of problems to be solved. After proving his first incompleteness theorem, Gödel had a suspicion that the continuum hypothesis may be independent of [[ZFC]]. He proved that the continuum hypothesis could not be disproved, by showing that it was true in his [[Inner model theory|inner model]] [[Constructible hierarchy|\(L\)]]. Cohen then proved it could not be proved either: given a countable standard transitive model \(M\) of ZFC, he proved that there was a forcing extension \(M[G]\) which added \(\aleph_2^M\) reals, and therefore that the continuum hypothesis fails within \(M[G]\). The powerful method of forcing could also be used to show the opposite: if \(M\) was a countable standard transitive model of ZFC, then there was a forcing extension \(M[G]\) that added a surjection from \(\aleph_1^M \to \mathfrak{c}^M\), and therefore the continuum hypothesis holds in \(M[G]\). As such, the continuum hypothesis is independent of ZFC, if it is consistent. It is therefore often regarded as one of the biggest unsolved problems in set theory. 740fdf9160787142f16b27003f4f2f01087fb02f 500 499 2023-09-03T18:07:29Z RhubarbJayde 25 wikitext text/x-wiki The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the [[axiom of choice]], and \(\aleph_1\) is the smallest [[Countability|uncountable]] cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the context of the [[axiom of determinacy]], it holds that, for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size, and yet \(2^{\aleph_0} \neq \aleph_1\) (in particular, the two are incomparable, since one needs choice to prove cardinals are linearly ordered). The continuum hypothesis was originally posed by Georg Cantor after his proof of [[Cantor's diagonal argument|the diagonal argument]]. Cantor was unable to prove or disprove the continuum hypothesis, primarily because he was unable to find a [[Well-ordered set|well-order]] of the reals. It is now known that there is no definable well-order on the reals: therefore, this approach can't work. Solving the continuum hypothesis was the first on Hilbert's famous list of problems to be solved in the 20th century. After proving his first incompleteness theorem, Gödel had a suspicion that the continuum hypothesis may be independent of [[ZFC]]. He proved that the continuum hypothesis could not be disproved, by showing that it was true in his [[Inner model theory|inner model]] [[Constructible hierarchy|\(L\)]]. Cohen then proved it could not be proved either: given a countable standard transitive model \(M\) of ZFC, he proved that there was a forcing extension \(M[G]\) which added \(\aleph_2^M\) reals, and therefore that the continuum hypothesis fails within \(M[G]\). The powerful method of forcing could also be used to show the opposite: if \(M\) was a countable standard transitive model of ZFC, then there was a forcing extension \(M[G]\) that added a surjection from \(\aleph_1^M \to \mathfrak{c}^M\), and therefore the continuum hypothesis holds in \(M[G]\). As such, the continuum hypothesis is independent of ZFC, if it is consistent. It is therefore often regarded as one of the biggest unsolved problems in set theory. b8e2038bf4b9c3bb88cd51c55f1a6899f7aea139 505 500 2023-09-04T20:19:10Z C7X 9 wikitext text/x-wiki The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the [[axiom of choice]], and \(\aleph_1\) is the smallest [[Countability|uncountable]] cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the context of the [[axiom of determinacy]], it holds that, for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size, and yet \(2^{\aleph_0} \neq \aleph_1\) (in particular, the two are incomparable, since one needs choice to prove cardinals are linearly ordered). The continuum hypothesis was originally posed by Georg Cantor after his proof of [[Cantor's diagonal argument|the diagonal argument]]. Cantor was unable to prove or disprove the continuum hypothesis, primarily because he was unable to find a [[Well-ordered set|well-order]] of the reals. It is now known that there is no definable well-order on the reals: therefore, this approach can't work. Solving the continuum hypothesis was the first on Hilbert's famous list of problems to be solved in the 20th century. After proving his first incompleteness theorem, Gödel had a suspicion that the continuum hypothesis may be independent of [[ZFC]]. He proved that the continuum hypothesis could not be disproved in ZFC, by showing that it was true in his [[Inner model theory|inner model]] [[Constructible hierarchy|\(L\)]].<ref>Any set theory text</ref> (Assuming ZFC is consistent to begin with, the alternative being that ZFC proves and refutes any statement by the principle of explosion.) Cohen then proved it could not be proved in ZFC either, assuming ZFC is consistent: given a countable standard transitive model \(M\) of ZFC, he proved that there was a forcing extension \(M[G]\) which added \(\aleph_2^M\) reals, and therefore that the continuum hypothesis fails within \(M[G]\).<ref>Any text about forcing</ref> The powerful method of forcing could also be used to show the opposite: if \(M\) was a countable standard transitive model of ZFC, then there was a forcing extension \(M[G]\) that added a surjection from \(\aleph_1^M \to \mathfrak{c}^M\), and therefore the continuum hypothesis holds in \(M[G]\). As such, the continuum hypothesis is independent of ZFC, if ZFC is consistent. It is therefore often regarded as one of the biggest unsolved problems in set theory. f5a65f4e884cb3c95aefbe318aa56b826ba486af 0 178 501 449 2023-09-04T11:07:13Z RhubarbJayde 25 wikitext text/x-wiki The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is trivial and intuitive. For example, one can see that the axiom of choice is equivalent to the assertion that the [[Cartesian product]] of any collection of nonempty sets is nonempty. Note that the assertion that the Cartesian product of finitely many nonempty sets is nonempty is obvious, but it's possible to define Cartesian product of infinitely many sets. Despite these simple characteristics, the axiom of choice is not a theorem of ZF, and it has some consequences that may be counterintuitive. For example, the axiom of choice is highly nonconstructive and doesn't actually tell somebody what that choice function looks like. Similarly, the axiom of choice tells us there is some well-order on the real numbers, but it is a theorem that there is no well-order on the real numbers. In general, the axiom of choice implies every set can be well-ordered: that is, for every set \(X\), there is a relation on \(X\) which imbues it with the structure necessary for it to be considered a [[well-ordered set]]. All finite sets, and even countable sets, can be trivially well-ordered, and in most cases this well-ordering will be definable, but the uncountable case is unclear. The proof that the axiom of choice implies that every set can be well-ordered is relatively simple. Namely, let \(Y\) be the family of subsets of \(X\). Let \(f\) be a choice function for \(Y\). Then define, via transfinite recursion, the [[ordinal]] indexed sequence \(a_\xi\) of elements of \(X\) by \(a_\xi = f(X \setminus \{a_\eta: \eta < \xi\})\). Every element of \(X\) shows up somewhere in this sequence. Therefore, define \(\leq\) by \(a_\xi \leq a_\eta\) iff \(\xi \leq \eta\). This is well-defined, and it is a well-order since the ordinals are well-ordered. Furthermore, the axiom of choice implies the law of excluded middle, which means constructivist mathematicians tend to work in ZF rather than ZFC. Lastly, and most famously, the axiom of choice implies the [[Banach-Tarski paradox]]. In particular, using the axiom of choice, it's possible to decompose any ball in 3D space into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. This is counterintuitive, but not truly paradoxical as the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. 8d6ed23c0710e26a925dea895d68bfecd3bc2c16 0 198 503 2023-09-04T11:23:29Z RhubarbJayde 25 Created page with "Hilbert's Grand Hotel is an analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the way infinite [[Bijection|bijections]] work and the fact that they go against common sense, it is possible to still fit many more people. Firstly, if there is a single new guest who wants a room, i..." wikitext text/x-wiki Hilbert's Grand Hotel is an analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the way infinite [[Bijection|bijections]] work and the fact that they go against common sense, it is possible to still fit many more people. Firstly, if there is a single new guest who wants a room, it is possible to accommodate them like so - namely, the hotel receptionist can tell everybody to move up one room, so the person checked into room zero moves to room one, the person checked into room one moves to room two, and so on. Because the set of rooms is never-ending, we don't run out of rooms and everybody who was checked in still has a room. Yet room number zero is now empty - the new guest can check in there. This is analogous to the proof that [[Omega|\(\omega\)]] and \(\omega+1\) are equinumerous. Similarly, if someone in room \(n\) checks out of the hotel, then everybody in room \(m\) for \(m > n\) can move one room to the left, and all the rooms will be filled again. 6a9df0f36a529ee4936d57e5431c594090a2ca04 0 199 504 2023-09-04T11:45:23Z RhubarbJayde 25 Created page with "The Banach-Tarski is a famous, counterintuitive consequence of the [[axiom of choice]]. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. The proof requires the axiom of choice, and, therefore, the truth of the Banach-Tars..." wikitext text/x-wiki The Banach-Tarski is a famous, counterintuitive consequence of the [[axiom of choice]]. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. The proof requires the axiom of choice, and, therefore, the truth of the Banach-Tarski paradox is a common argument against the usage of the axiom of choice. f7179f8eba8bc50de9fafe9b845ba410686dc8fe 1 200 506 2023-09-04T20:20:32Z C7X 9 /* "unsolved problems of set theory" */ new section wikitext text/x-wiki == "unsolved problems of set theory" == Does this mean "unsolved" as in open, or "unsolved" as in unsolvable (both problems of finding a proof from ZFC of CH or finding a proof from ZFC of !CH are impossible)? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:20, 4 September 2023 (UTC) c1b43b4f9284cfa4775d97ec3372b6d9420d2d6c 0 121 507 334 2023-09-04T21:15:43Z C7X 9 wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>https://arxiv.org/abs/1708.06669</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved both the truth of the reflection principle over \(\mathrm{ZF}\).<ref></ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-correct. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give sine informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. 78a11954a61553dcb7df54438a7baeb894cb9e21 508 507 2023-09-04T21:17:15Z C7X 9 wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>https://arxiv.org/abs/1708.06669</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ;''he Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-correct. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give sine informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. 6bca9e44ea8d0cca43408747a7b89e3d97aff6fb 509 508 2023-09-04T21:17:56Z C7X 9 wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>https://arxiv.org/abs/1708.06669</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-correct. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give sine informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. ==References== <reflist /> eb50f12b283c31ac068c2d34f79fe1c8bde6c946 510 509 2023-09-04T21:18:04Z C7X 9 /* References */ wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>https://arxiv.org/abs/1708.06669</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-correct. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give sine informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. ==References== cbf093c9f8f791a4ac3f92fb87a200cbed492a36 3 201 511 2023-09-04T22:39:35Z Augigogigi 2 /* The Forum */ new section wikitext text/x-wiki == The Forum == Anything i can do to help / speed things along? → [[User:Augigogigi|<span style="color:#008080;">Augi</span>]] 22:39, 4 September 2023 (UTC) 686814de014cb15191eb7ddb8d7201f540069025 2 2 512 136 2023-09-04T22:40:06Z Augigogigi 2 wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. 926cd2b8582290cf8ccf222d6ff3d75fde095f12 0 202 514 2023-09-06T18:18:57Z RhubarbJayde 25 Created page with "The cofinality of an [[ordinal]] \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's countabl..." wikitext text/x-wiki The cofinality of an [[ordinal]] \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's [[countable]], then it's exactly \(\omega\). It is easy to see that \(\mathrm{cof}(\alpha) \leq \alpha\) for all \(\alpha\), because the identity has unbounded range. Also, \(\mathrm{cof}(\mathrm{cof}(\alpha)) = \mathrm{cof}(\alpha)\), because if there is a \(\delta < \mathrm{cof}(\alpha)\) and maps \(f: \delta \to \mathrm{cof}(\alpha)\), \(g: \mathrm{cof}(\alpha) \to \alpha\) with unbounded range, then \(g \circ f: \delta \to \alpha\) also has unbounded range, contradicting minimality of \(\mathrm{cof}(\alpha)\). An ordinal is regular if it is equal to its own cofinality, else it is singular. So: * [[0]], [[1]] and [[Omega|\(\omega\)]] are regular. * All natural numbers other than \(1\) are singular. * All countable ordinals other than \(\omega\) are singular. * \(\mathrm{cof}(\alpha)\) is regular for any \(\alpha\). Cofinality is used in the definition of [[Inaccessible cardinal|weakly inaccessible]] cardinals. f99a9d61f5d79a3031b7afba8ab1b518d9c98e6f 0 105 515 496 2023-09-06T18:19:26Z RhubarbJayde 25 wikitext text/x-wiki There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first introduced by Hausdorff in an attempt to resolve the [[continuum hypothesis]].<ref>Hausdorff, Felix. "[https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:%5b453%5d} Grundzüge einer Theorie der geordneten Mengen]", Mathematische Annalen, vol. 65, num. 4 (1908), pp.435--505. DOI:10.1007/BF01451165.</ref> Cardinals \( \aleph_\alpha \) for limit ordinal \( \alpha \) are known as limit cardinals<ref>Jech, Thomas (2003), ''Set Theory'', Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag</ref>, since applying the cardinal successor operator to a cardinal less than \( \aleph_\alpha \) yields a cardinal also less than \( \aleph_\alpha \). However, \( \aleph_\omega \) and many other limit cardinals have a short cofinal sequence, these cardinals are called [[Cofinality|singular]]. Formally, \( \kappa \) is singular if there is some sequence of \( < \kappa \) smaller ordinals whose limit is \( \kappa \). For example, \( \aleph_\omega \) is the limit of the sequence \( \aleph_0 \), \( \aleph_1 \), \( \aleph_2 \), ... which has length \( < \aleph_\omega \). Meanwhile, for cardinals such as \( \aleph_1 \), the limit of any countable sequence of countable ordinals is countable, so \( \aleph_1 \) is not singular - aka [[. We now call a cardinal \( \kappa \) weakly inaccessible if it is regular ''and'' a limit cardinal. You can see that if \( \kappa \) is weakly inaccessible and \( \alpha < \kappa \) then \( \aleph_\alpha < \kappa \) too, and then regularity gives that the limit of the length-\( \omega \) sequence \( \alpha \), \( \aleph_\alpha \), \( \aleph_{\aleph_\alpha} \), ... is less than \( \kappa \) as well. The existence of weakly inaccessible cardinals is - surprisingly - not provable in \( \mathrm{ZFC} \), assuming its existence. We explain why in the third section. However, authors may typically assume their existence and use them in ordinal collapsing functions to describe ordinals equal to or greater than the [[Extended Buchholz ordinal|EBO]], since an inaccessible cardinal acts as a suitable "diagonalizer" over \( \alpha \mapsto \Omega_\alpha \), like how \( \Omega \) acts as a suitable diagonalizer in an ordinal collapsing function such as Madore's or Bachmann's. == Strongly inaccessible == Strongly inaccessible cardinals were defined later, but are more common than weakly inaccessible cardinals. In particular, the term "inaccessible" typically refers to strongly inaccessible while in apeirological circles, it typically refers to weakly inaccessible cardinals. They are defined as a subtle generalization of weakly inaccessible cardinals - instead of requiring that, for all \( \lambda < \kappa \), we have \( \lambda^+ < \kappa \) (and thus also \( \lambda^{+(n)} < \kappa \) for all \( n < \omega \)), we require \( 2^\lambda < \kappa \). Note that here \( 2^\lambda \) refers to the notions of cardinal arithmetic, instead of ordinal arithmetic, and is equal to the cardinality of \( \mathcal{P}(\lambda) \), which is always greater than \( \lambda \) by a theorem of Cantor. This is unlike the fact that if \( \lambda \) is a cardinal (and thus [[Epsilon numbers|epsilon number]]), we have \( 2^\lambda = \lambda \) w.r.t. ordinal arithmetic. Intuitively, \( \kappa \) is completely unreachable from below by using replacement, powersets and limits. This explains the naming. Notice that if the uncountability requirement is dropped, \( \aleph_0 \) is actually strongly inaccessible and weakly inaccessible! This is because that the limit of any finite sequence of natural numbers is finite, and that if \( n \) is finite, so are \( n^+ = n + 1 \) and \( 2^n \). Zermelo referred to the strongly inaccessible cardinals including \( \aleph_0 \) as "Grenzzahlen".<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref> However, \( \aleph_0 \) is usually not considered strongly inaccessible. Neither is zero, although it similarly would be weakly inaccessible, although vacuously. For this reason many authors add the condition of uncountability. Thus, like how \( \aleph_0 \) is infinite and transcends the notion of finiteness, inaccessible cardinals transcend smaller cardinals. This is why [[Large cardinal|large cardinals]] are often called the "higher infinite". You can easily see that any strongly inaccessible cardinal is weakly inaccessible as well. This is because if \( \lambda < \kappa \) then \( 2^\lambda < \kappa \) and \( \lambda^+ \leq 2^\lambda \), thus \( \lambda^+ < \kappa \) as well. However, assuming axioms such as the resurrection axiom, it is possible for \( 2^{\aleph_0} \) to be weakly inaccessible, while not being strongly inaccessible by any means.<ref>Resurrection axiom paper</ref> == Grothendieck universes and categoricity == Alexander Grothendieck, a famous algebraic geometrist, introduced the notion of a Grothendieck universe, a large set in which essential set theory could be performed, to avoid having to work in second-order set theory and have to deal with proper classes, in his work. They are transitive, closed under powerset, closed under strong unions, and more. However, ironically, the only instances of Grothendieck universes which can be proven to exist in ZFC, are the empty set and \( V_\omega \) in the von Neumann hierarchy. These clearly don't do, since they don't contain, say the set of real numbers. To avoid having to go beyond \( \mathrm{ZFC} \), he introduced a notion that required you to go beyond \( \mathrm{ZFC} \)! And any Grothendieck universe has to be of the form \( V_\kappa \), in the von Neumann hierarchy, where \( \kappa = 0 \), \( \kappa = \omega \), or \( \kappa \) is strongly inaccessible. In other words, because of how inaccessible a strongly inaccessible cardinal is from below in terms of cardinal operations and transfinite recursion, the collection of sets with rank \( < \kappa \) forms an ideal model of the universe of set theory. A similar result is Zermelo's famous second-order categoricity theorem. Zermelo proved that, for any cardinal \( \kappa \), the rank \( V_\kappa \) satisfies Morse-Kelley set theory (a powerful, second-order extension of \( \mathrm{ZFC} \)) if, and only if, \( \kappa = 0 \) or \( \kappa \) is strongly inaccessible. This helps us to prove why weakly and strongly inacccessible cardinals can't exist in \( \mathrm{ZFC} \), assuming its consistency. Gödel's second incompleteness theorem tells us that, if \( \mathrm{ZFC} \) is consistent, then it can't prove that it's consistent, and thus Gödel's completeness theorem tells us that being consistent is equivalent to having a model. Since \( V_\kappa \) satisfies Morse-Kelley set theory, and thus \( \mathrm{ZFC} \), whenever \( \kappa \) is strongly inaccessible, \( \mathrm{ZFC} \) can't prove there is a strongly inaccessible cardinal, if it's consistent. And now, assuming \( \mathrm{GCH} \), we have \( \lambda^+ = 2^\lambda \) for all \( \lambda \), and thus \( \kappa \) is strongly inaccessible iff it is weakly inaccessible. Since \( \mathrm{ZFC} + \mathrm{GCH} \) can't prove the existence of strongly inaccessible cardinals, it can't prove the existence of weakly inaccessible cardinals either. And anything provable in \( \mathrm{ZFC} \) is provable in \( \mathrm{ZFC} + \mathrm{GCH} \) as well, hence weakly inaccessible cardinals don't necessarily exist. 155115e8b4ac59c8733a40bb9c211c626f245978 0 203 516 2023-09-07T18:42:33Z RhubarbJayde 25 Redirected page to [[Filter]] wikitext text/x-wiki #REDIRECT [[Filter]] 6101580ef9d0062723e8e84c6e9fe4bbcfe658e2 0 204 517 2023-09-07T19:12:30Z RhubarbJayde 25 Created page with "A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A..." wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a maximal filter: given some \(x \in X\), the maximal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). It is also easy to see this is a filter, however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a cardinal \(\gamma\) and set \(X\), a filter \(F\) on \([X]^{< \gamma}\) is called fine if, for each \(x \in X\), \(\{\sigma \in [X]^{< \gamma}: x \in \sigma\} \in F\). Similarly, an ultrafilter on a cardinal \(\kappa\) is called fine (or uniform) if, for each \(\alpha < \kappa\), \(\{\sigma < \kappa: \alpha < \sigma\} = (\sigma,\kappa) \in F\). e22a0eb391261d0158a47b9d9e86620b6ef230ab 518 517 2023-09-07T19:13:19Z RhubarbJayde 25 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a maximal filter: given some \(x \in X\), the maximal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). It is also easy to see this is a filter, however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a cardinal \(\gamma\) and set \(X\), a filter \(F\) on \([X]^{< \gamma}\) is called fine if, for each \(x \in X\), \(\{\sigma \in [X]^{< \gamma}: x \in \sigma\} \in F\). Similarly, an ultrafilter on a cardinal \(\kappa\) is called fine (or uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). 3cd7ae4b6c1fa97397c8aa81a86ae3b74c62c4ce 519 518 2023-09-07T23:20:34Z C7X 9 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a maximal filter: given some \(x \in X\), the maximal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore studies nonprincipal (possibly definition needed) filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). It is also easy to see this is a filter, however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a cardinal \(\gamma\) and set \(X\), a filter \(F\) on \([X]^{< \gamma}\) is called fine if, for each \(x \in X\), \(\{\sigma \in [X]^{< \gamma}: x \in \sigma\} \in F\). Similarly, an ultrafilter on a cardinal \(\kappa\) is called fine (or uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). d105ce849728034700a1120fc43a7ad588ce94cd 527 519 2023-09-08T16:04:39Z RhubarbJayde 25 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a maximal filter: given some \(x \in X\), the principal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore typically studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). It is also easy to see this is a filter, however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). Note that, if \(X = \bigcup X\), then every fine filter is nonprincipal: assume \(F\) is principal, witnessed by \(x \in X\). Then \(x \in \bigcup X\) so, if \(F\) were fine, then \(\{s \in X: x \in s\} \in F\), thus \(x \in x\) - a contradiction! Similarly, a filter \(F\) on \(X\) is called normal, if, for each function \(f: X \to \bigcup X\), if \(\{s \in X: f(s) \in s\} \in F\), then there is some \(x \in \bigcup X\) so that \(\{s \in X: f(s) = x\} \in F\). This definition is inspired by the [[Fodor's lemma|pressing-down lemma]]. Dually to the fact that no fine filter is principal, every principal filter is normal: assume \(F\) is principal, witnessed by \(x \in X\), and \(f: X \to \bigcup X\). Then, if \(\{s \in X: f(s) \in s\} \in F\), we have \(f(x) \in x\), and so, letting \(x' = f(x)\), we have \(x' \in \bigcup X\), and \(\{s \in X: f(s) = x'\} \in F\). Note, however, that there can be nonprincipal filters which are normal - it is even possible for a filter to be both normal and fine! If \(F\) is a fine filter on \(X\), and \(X\) is a [[well-ordered set]], then all elements of \(F\) have the same cardinality as \(X\), and therefore fine (or even normal fine) ultrafilters may more closely approximate the notion of largeness - assume \(Y \in F\) and \(|Y| < |X|\). Let \(\leq\) be the well-order on \(X\), and let \(x\) be the \(\leq\)-least element of \(X\) which is greater than all elements of \(Y\). By fineness, we have \(\{s \in X: x \in s\} \in F\), and therefore \(\{s \in X: x \in s\} \cap X \in F\) since \(F\) is a filter. However, \(\{s \in X: x \in s\} \cap X = \emptyset\), contradicting \(\emptyset \notin F\). 1d8f35865d98da414c0b397e691b9a3f373df483 528 527 2023-09-08T17:14:26Z RhubarbJayde 25 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a maximal filter: given some \(x \in X\), the principal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore typically studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). It is also easy to see this is a filter, however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). Note that, if \(X \subseteq \bigcup X\), then every fine filter is nonprincipal: assume \(F\) is principal, witnessed by \(x \in X\). Then \(x \in \bigcup X\) so, if \(F\) were fine, then \(\{s \in X: x \in s\} \in F\), thus \(x \in x\) - a contradiction! Similarly, a filter \(F\) on \(X\) is called normal, if, for each function \(f: X \to \bigcup X\), if \(\{s \in X: f(s) \in s\} \in F\), then there is some \(x \in \bigcup X\) so that \(\{s \in X: f(s) = x\} \in F\). This definition is inspired by the [[Fodor's lemma|pressing-down lemma]]. Dually to the fact that no fine filter is principal, every principal filter is normal: assume \(F\) is principal, witnessed by \(x \in X\), and \(f: X \to \bigcup X\). Then, if \(\{s \in X: f(s) \in s\} \in F\), we have \(f(x) \in x\), and so, letting \(x' = f(x)\), we have \(x' \in \bigcup X\), and \(\{s \in X: f(s) = x'\} \in F\). Note, however, that there can be nonprincipal filters which are normal - it is even possible for a filter to be both normal and fine! If \(F\) is a fine filter on \(X\), and \(X\) is a [[well-ordered set]], then all elements of \(F\) have the same cardinality as \(X\), and therefore fine (or even normal fine) ultrafilters may more closely approximate the notion of largeness - assume \(Y \in F\) and \(|Y| < |X|\). Let \(\leq\) be the well-order on \(X\), and let \(x\) be the \(\leq\)-least element of \(X\) which is greater than all elements of \(Y\). By fineness, we have \(\{s \in X: x \in s\} \in F\), and therefore \(\{s \in X: x \in s\} \cap X \in F\) since \(F\) is a filter. However, \(\{s \in X: x \in s\} \cap X = \emptyset\), contradicting \(\emptyset \notin F\). b615144e7202a2fa6fe182da117d72947e1198e0 561 528 2023-09-09T16:43:01Z RhubarbJayde 25 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a principal filter: given some \(x \in X\), the principal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore typically studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). It is also easy to see this is a filter, however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). Note that, if \(X \subseteq \bigcup X\), then every fine filter is nonprincipal: assume \(F\) is principal, witnessed by \(x \in X\). Then \(x \in \bigcup X\) so, if \(F\) were fine, then \(\{s \in X: x \in s\} \in F\), thus \(x \in x\) - a contradiction! Similarly, a filter \(F\) on \(X\) is called normal, if, for each function \(f: X \to \bigcup X\), if \(\{s \in X: f(s) \in s\} \in F\), then there is some \(x \in \bigcup X\) so that \(\{s \in X: f(s) = x\} \in F\). This definition is inspired by the [[Fodor's lemma|pressing-down lemma]]. Dually to the fact that no fine filter is principal, every principal filter is normal: assume \(F\) is principal, witnessed by \(x \in X\), and \(f: X \to \bigcup X\). Then, if \(\{s \in X: f(s) \in s\} \in F\), we have \(f(x) \in x\), and so, letting \(x' = f(x)\), we have \(x' \in \bigcup X\), and \(\{s \in X: f(s) = x'\} \in F\). Note, however, that there can be nonprincipal filters which are normal - it is even possible for a filter to be both normal and fine! If \(F\) is a fine filter on \(X\), and \(X\) is a [[well-ordered set]], then all elements of \(F\) have the same cardinality as \(X\), and therefore fine (or even normal fine) ultrafilters may more closely approximate the notion of largeness - assume \(Y \in F\) and \(|Y| < |X|\). Let \(\leq\) be the well-order on \(X\), and let \(x\) be the \(\leq\)-least element of \(X\) which is greater than all elements of \(Y\). By fineness, we have \(\{s \in X: x \in s\} \in F\), and therefore \(\{s \in X: x \in s\} \cap X \in F\) since \(F\) is a filter. However, \(\{s \in X: x \in s\} \cap X = \emptyset\), contradicting \(\emptyset \notin F\). 8b793ed1b609f1877f8351845250904b5221f2eb 562 561 2023-09-09T16:45:38Z RhubarbJayde 25 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a principal filter: given some \(x \in X\), the principal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore typically studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). All sets in the Fréchet filter are intuitively large since they must have cardinality \(\kappa\) - it is also easy to see this is in fact a filter and not a misnomer; however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). Note that, if \(X \subseteq \bigcup X\), then every fine filter is nonprincipal: assume \(F\) is principal, witnessed by \(x \in X\). Then \(x \in \bigcup X\) so, if \(F\) were fine, then \(\{s \in X: x \in s\} \in F\), thus \(x \in x\) - a contradiction! Similarly, a filter \(F\) on \(X\) is called normal, if, for each function \(f: X \to \bigcup X\), if \(\{s \in X: f(s) \in s\} \in F\), then there is some \(x \in \bigcup X\) so that \(\{s \in X: f(s) = x\} \in F\). This definition is inspired by the [[Fodor's lemma|pressing-down lemma]]. Dually to the fact that no fine filter is principal, every principal filter is normal: assume \(F\) is principal, witnessed by \(x \in X\), and \(f: X \to \bigcup X\). Then, if \(\{s \in X: f(s) \in s\} \in F\), we have \(f(x) \in x\), and so, letting \(x' = f(x)\), we have \(x' \in \bigcup X\), and \(\{s \in X: f(s) = x'\} \in F\). Note, however, that there can be nonprincipal filters which are normal - it is even possible for a filter to be both normal and fine! If \(F\) is a fine filter on \(X\), and \(X\) is a [[well-ordered set]], then all elements of \(F\) have the same cardinality as \(X\), and therefore fine (or even normal fine) ultrafilters may more closely approximate the notion of largeness - assume \(Y \in F\) and \(|Y| < |X|\). Let \(\leq\) be the well-order on \(X\), and let \(x\) be the \(\leq\)-least element of \(X\) which is greater than all elements of \(Y\). By fineness, we have \(\{s \in X: x \in s\} \in F\), and therefore \(\{s \in X: x \in s\} \cap X \in F\) since \(F\) is a filter. However, \(\{s \in X: x \in s\} \cap X = \emptyset\), contradicting \(\emptyset \notin F\). 3d75010745cd29e71793cf9154549b7093c554cf 563 562 2023-09-09T16:48:09Z RhubarbJayde 25 wikitext text/x-wiki A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above [[Measurable|measurable cardinals]], although they also have some relation to [[Reflection principle|indescribable]] and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A \in F\) and \(A \subseteq B\) then \(B \in F\). Intuitively, a filter is a predicate which chooses out exactly which subsets of \(X\) are large; namely: * \(X\) is large. * The [[empty set]] is not large. * The intersection of large sets is large. * A superset of a large set is large. An ultrafilter is a maximal filter: for every subset \(Y\) of \(X\), either \(Y\) is large or its complement (\(X \setminus Y\)) is large. The reason these are maximal is, because if \(F' \supset F\), then there is some \(Y \in F'\) so that \(Y \notin F\). Therefore \(X \setminus Y \in F\), and so \(X \setminus Y \in F'\), therefore \((X \setminus Y) \cap Y = \emptyset \in F'\), so \(F'\) can't be a filter. One of the most natural examples of a filter is a principal filter: given some \(x \in X\), the principal filter with respect to \(x\) has \(Y\) large iff \(x \in Y\). It is easy to verify this is an ultrafilter, however this doesn't exactly match with one's intuition of largeness, since even singletons can be large in this interpretation. One therefore typically studies nonprincipal filters instead - another easy example is the Fréchet filter, the filter on a cardinal \(\kappa\) defined by \(X\) being large iff \(|\kappa \setminus X| < \kappa\). All sets in the Fréchet filter are intuitively large since they must have cardinality \(\kappa\) - it is also easy to see this is in fact a filter and not a misnomer; however, it is not an ultrafilter. It is possible to impose further conditions, other than the four in the definition of the filter and nonprincipality. This includes \(\gamma\)-completeness, for a cardinal \(\gamma\), which asserts that the filter is closed not just intersection of two sets, but of \(< \gamma\)-many sets. Note that any filter is \(\omega\)-complete. A cardinal \(\kappa\) with a \(\kappa\)-complete ultrafilter on \(\kappa\) is precisely a [[measurable]] cardinal, and thus the existence of such a cardinal is unprovable in [[ZFC]], assuming its consistency. For a set \(X\), a filter \(F\) on \(X\) is called fine if, for each \(x \in \bigcup X\), \(\{s \in X: x \in s\} \in F\). In particular, an ultrafilter on a cardinal \(\kappa\) is fine (in this case, also known as uniform) if, for each \(\alpha < \kappa\), \(\{\sigma: \alpha < \sigma < \kappa\} \in F\). This also makes sense, since "almost all" ordinals below \(\kappa\) will be in this interval. If \(F\) is a fine filter on \(X\), and \(X\) is a [[well-ordered set]], then all elements of \(F\) have the same cardinality as \(X\), and therefore fine (or even normal fine) ultrafilters may more closely approximate the notion of largeness - assume \(Y \in F\) and \(|Y| < |X|\). Let \(\leq\) be the well-order on \(X\), and let \(x\) be the \(\leq\)-least element of \(X\) which is greater than all elements of \(Y\). By fineness, we have \(\{s \in X: x \in s\} \in F\), and therefore \(\{s \in X: x \in s\} \cap X \in F\) since \(F\) is a filter. However, \(\{s \in X: x \in s\} \cap X = \emptyset\), contradicting \(\emptyset \notin F\). Note that, if \(X \subseteq \bigcup X\), then every fine filter is nonprincipal: assume \(F\) is principal, witnessed by \(x \in X\). Then \(x \in \bigcup X\) so, if \(F\) were fine, then \(\{s \in X: x \in s\} \in F\), thus \(x \in x\) - a contradiction! Similarly, a filter \(F\) on \(X\) is called normal, if, for each function \(f: X \to \bigcup X\), if \(\{s \in X: f(s) \in s\} \in F\), then there is some \(x \in \bigcup X\) so that \(\{s \in X: f(s) = x\} \in F\). This definition is inspired by the [[Fodor's lemma|pressing-down lemma]]. Dually to the fact that no fine filter is principal, every principal filter is normal: assume \(F\) is principal, witnessed by \(x \in X\), and \(f: X \to \bigcup X\). Then, if \(\{s \in X: f(s) \in s\} \in F\), we have \(f(x) \in x\), and so, letting \(x' = f(x)\), we have \(x' \in \bigcup X\), and \(\{s \in X: f(s) = x'\} \in F\). Note, however, that there can be nonprincipal filters which are normal - it is even possible for a filter to be both normal and fine! 2a0e3d9c04afc3952881ede1a4e5fd7ac8fddaff 0 69 520 332 2023-09-07T23:22:00Z C7X 9 /* Stability */ wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref name="OrdinalArithmeticSigmaOne">T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997). Accessed 29 August 2023.</ref>^{implicit in section 3}<ref name="ElementaryPatterns" />^{corollary 6.12}<ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "Patterns of resemblance of order two", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Reflection criterion== Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns. (I think <ref name="OrdinalArithmeticSigmaOne" /> is a citation) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. ==Citations== 0f7dd81e85252ad904236c0225afec9db6d85b33 521 520 2023-09-07T23:23:53Z C7X 9 wikitext text/x-wiki The patterns of resemblance (PoR) are a system of ordinal-notations introduced by T. J. Carlson. Like the notion of stability for ordinals it uses elementary substructures, however between ordinals themselves, instead of between ranks of the constructible universe. Carlson's \(<_n\)-relations have a property known as the respecting property, which also holds for the \(\Sigma_n\)-relations between ranks of \(L\), and for parenthood relations in [[Bashicu matrix system|BMS version 4]]. For this reason, pure patterns of resemblance were originally believed to have the same limit of representable ordinals as BMS. A pattern is known as isominimal if it is pointwise least among all patterns isomorphic to it. The ''core'' is the set of ordinals which occur in an isominimal pattern.<ref name="ElementaryPatterns">T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007200000403 Elementary Patterns of Resemblance]" (2001). Annals of Pure and Applied Logic vol. 108, pp.19--77.</ref> The definition of the core depends on which system is used, and as there are different systems going by the name "patterns of resemblance" (such as pure second-order patterns and additive first-order patterns), the term "the core" is context-dependent, and is defined analogously.<ref name="PureSigma2Beyond">G. Wilken, "[https://arxiv.org/abs/1710.01870v5 Pure \(\Sigma_2\)-Elementarity beyond the Core]" (2021), p.6. Accessed 29 August 2023.</ref> For all systems currently analyzed, the core is a recursive ordinal.<ref name="OrdinalArithmeticSigmaOne">T. J. Carlson, "[https://www.researchgate.net/publication/257334588_Ordinal_arithmetic_and_Sigma_1-elementarity Ordinal Arithmetic and \(\Sigma_1\)-Elementarity]" (1997). Accessed 29 August 2023.</ref>^{implicit in section 3}<ref name="ElementaryPatterns" />^{corollary 6.12}<ref name="PurePatternsOrderTwo">G. Wilken, "[https://arxiv.org/abs/1608.08421v5 Pure patterns of order 2]", corollary 4.10. Annals of Pure and Applied Logic vol. 169 (2018), pp.54--82.</ref><ref>T. J. Carlson, "[https://www.sciencedirect.com/science/article/pii/S0168007208001760 Patterns of resemblance of order two]", corollary 15.15. Annals of Pure and Applied Logic vol. 158 (2009), pp.90--124.</ref> A characterization of the core for additive second-order patterns is not currently known, but Wilken expects that it is equal to the proof-theoretic ordinal of KPI (axiomatization of "admissible limits of admissible universes").<ref name="PurePatternsOrderTwo" />^p.23<ref name="PureSigma2Beyond" />^p.6 (Although the second source claims that this ordinal is obtained from collapsing \(\omega\)-many weakly inaccessible cardinals - check this) ==Reflection criterion== Let \(a \subseteq_{fin} b\) hold iff \(a\) is a finite subset of \(b\), and use interval notation for ordinals. \(\alpha <_1 \beta\) holds iff for all \(X \subseteq_{fin} [0,\alpha)\) and \(Y \subseteq_{fin} [\alpha,\beta)\), there exists a \(\tilde Y\subseteq_{fin} [0,\alpha)\) such that \(X \cup Y \cong X \cup \tilde Y\), where \(\cong\) is isomorphism with respect to the language of first-order patterns. (I think <ref name="OrdinalArithmeticSigmaOne" /> is a citation) ==Stability== It has been known since (Carlson 2001) that certain variants of patterns of resemblance involving stability result in a core isomorphic to that of the usual patterns of resemblance. In particular, if \(\alpha\preceq\beta\) is interpreted as \(L_\alpha\prec_{\Sigma_1}L_\beta\), then the core of \((\textrm{Ord},0,+,\leq,\preceq)\) is isomorphic to the core of additive first-order patterns,<ref name="ElementaryPatterns" />^p.20 so it has order type \(\psi_0(\Omega_\omega)\). This may be seen as somewhat similar to the connection between BMS and stability used in Yto's termination proof for BMS. ==Citations== 43f1b774f824e90b9c0de8065df351ca0acd2c31 0 202 522 514 2023-09-07T23:24:52Z C7X 9 wikitext text/x-wiki The cofinality of an [[ordinal]] \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's [[countable]], then it's exactly \(\omega\). It is easy to see that \(\mathrm{cof}(\alpha) \leq \alpha\) for all \(\alpha\), because the identity has unbounded range. Also, \(\mathrm{cof}(\mathrm{cof}(\alpha)) = \mathrm{cof}(\alpha)\), because if there is a \(\delta < \mathrm{cof}(\alpha)\) and maps \(f: \delta \to \mathrm{cof}(\alpha)\), \(g: \mathrm{cof}(\alpha) \to \alpha\) with unbounded range, then \(g \circ f: \delta \to \alpha\) also has unbounded range, contradicting minimality of \(\mathrm{cof}(\alpha)\). An ordinal is regular if it is equal to its own cofinality, else it is singular. So: * [[0]], [[1]] and [[Omega|\(\omega\)]] are regular. * All natural numbers other than \(1\) are singular. * All countable ordinals other than \(\omega\) are singular. * \(\mathrm{cof}(\alpha)\) is regular for any \(\alpha\). Cofinality is used in the definition of [[Inaccessible cardinal|weakly inaccessible]] cardinals. ==Without choice== Citation about every uncountable cardinal being singular being consistent with ZF 94c46cca8d748495b949331bf4b481b1fa47d675 1 205 523 2023-09-07T23:34:37Z C7X 9 /* Naturality of CH */ new section wikitext text/x-wiki == Naturality of CH == Many mathematicians seem to doubt that CH is a natural question, an extreme quote from this position is by Folland: : "My own feeling, subject to revision in the event of a major breakthrough in set theory, is that if the answer to one's question turns out to depend on the continuum hypothesis, one should give up and ask a different question." ([https://cs.nyu.edu/pipermail/fom/2000-June/004041.html]) IIRC the general perception of CH as unnatural is what led Friedman to begin looking for natural statements independent of ZFC (if CH were widely viewed as natural, there would already be a known natural statement independent of ZFC, and his work would be done) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 23:34, 7 September 2023 (UTC) fe75a52aac199b063c9c6fce7d92b8bd3dbc5ad2 0 36 524 357 2023-09-07T23:37:05Z C7X 9 wikitext text/x-wiki A [[set]] is said to be '''finite''' if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be [[infinite]]. More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a [https://en.wikipedia.org/wiki/Bijection bijective function] (a one-to-one correspondence) \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). In terms of von Neumann ordinals, this is equivalent to there being some well-ordering on the set whose order-type is finite. The unique natural for which this holds is called its [[cardinality]], although this concept may be defined in greater generality. Perhaps the simplest finite set is the [[empty set]] \(\varnothing\), whose cardinality is [[0]]. Any singleton set \(\{a\}\) is finite and has cardinality [[1]]. An [[ordinal]] is called '''finite''' when it's the [[order type]] of a finite [[well-ordered set]]. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers. Likewise, a [[cardinal]] is called '''finite''' when it's the cardinality of a finite set. Once again, finite cardinals can be identified with the natural numbers. == Dedekind finiteness == Dedekind has the following definition of finiteness that does not make reference to \(\mathbb N\): : \(X\) is finite if there is no proper subset of \(X\) that has the same cardinality as \(X\). Without the axiom of choice it cannot be proven that this is equivalent to the \(\mathbb N\)-based definition of finiteness.{{citation needed}} == Properties == * Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite. * The powerset of a finite set is finite. * The union of two finite sets, and thus of finitely many finite sets, is finite. * The [[ordinal sum|sum]], [[ordinal product|product]], or [[ordinal exponentiation|exponentiation]] of two finite ordinals is finite. * The [[cardinal sum|sum]], [[cardinal product|product]], or [[cardinal exponentiation|exponentiation]] of two finite cardinals is finite. == External links == * {{Mathworld|Finite Set|author=Barile, Margherita}} * {{Wikipedia|Finite set}} 9fab8addf92da1f664620dd0695927233c6e876e 1 206 525 2023-09-08T01:57:43Z C7X 9 /* "Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory" */ new section wikitext text/x-wiki == "Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory" == Since \(\beta_0\) can be defined in the language of first-order set theory I think there is an injection from it to \(\mathbb N\). Since \(\beta_0\) is in \(L_{\textrm{least stable ordinal}}\), by theorem 7.8 of chapter V of Marek's "[http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and related facts]" (Fundamenta Mathematicae vol. 82, 1974), there is a \(\Sigma_1\) definition of \(\beta_0\) in \(V\) (i.e. there is a \(\Sigma_1\) formula \(\phi(x)\) such that \(\forall x(\phi(x)\leftrightarrow x=\beta_0)\)). Every ordinal \(<\beta_0\) has a \(\Sigma_1\) definition in \(V\) as well. Since satisfaction of \(\Sigma_1\) formulae in \(V\) is definable, the map that takes an ordinal \(\alpha<\beta_0\) to the smallest Gödel-codes of a \(\Sigma_1\) formula defining \(\alpha\) is a definable map. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 01:57, 8 September 2023 (UTC) da2d8ec3fc8378d2d01f18d0f7599e7b7b74b2bd 526 525 2023-09-08T01:57:59Z C7X 9 Typo /* "Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory" */ wikitext text/x-wiki == "Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory" == Since \(\beta_0\) can be defined in the language of first-order set theory I think there is an injection from it to \(\mathbb N\). Since \(\beta_0\) is in \(L_{\textrm{least stable ordinal}}\), by theorem 7.8 of chapter V of Marek's "[http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and related facts]" (Fundamenta Mathematicae vol. 82, 1974), there is a \(\Sigma_1\) definition of \(\beta_0\) in \(V\) (i.e. there is a \(\Sigma_1\) formula \(\phi(x)\) such that \(\forall x(\phi(x)\leftrightarrow x=\beta_0)\)). Every ordinal \(<\beta_0\) has a \(\Sigma_1\) definition in \(V\) as well. Since satisfaction of \(\Sigma_1\) formulae in \(V\) is definable, the map that takes an ordinal \(\alpha<\beta_0\) to the smallest Gödel-code of a \(\Sigma_1\) formula defining \(\alpha\) is a definable map. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 01:57, 8 September 2023 (UTC) e46d6da5470b65f21699f9c6329093c7cd41159c 0 207 529 2023-09-08T17:38:18Z RhubarbJayde 25 Created page with "Supercompact cardinals are a kind of [[large cardinal]] with powerful reflection properties. The construction of an inner model accommodating a supercompact cardinal is considered the holy grail of [[inner model theory]], and is extremely difficult. Formally, a cardinal \(\kappa\) is called \(\lambda\)-supercompact iff there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) a..." wikitext text/x-wiki Supercompact cardinals are a kind of [[large cardinal]] with powerful reflection properties. The construction of an inner model accommodating a supercompact cardinal is considered the holy grail of [[inner model theory]], and is extremely difficult. Formally, a cardinal \(\kappa\) is called \(\lambda\)-supercompact iff there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). In particular, any measurable cardinal is \(\kappa\)-supercompact, and any \(\gamma\)-supercompact cardinal is \(\gamma\)-strong. Like how measurable cardinals can alternatively be defined in terms of ultrafilters, so can supercompact cardinals: \(\kappa\) is \(\lambda\)-supercompact iff there is a \(\kappa\)-complete, normal, fine [[Filter|ultrafilter]] on \([\lambda]^{< \kappa}\), the set of subsets of \(\lambda\) with size less than \(\kappa\). Then \(\kappa\) is supercompact iff it is \(\lambda\)-supercompact for all \(\lambda > \kappa\). As previously mentioned, any supercompact cardinal is strong (and therefore has Mitchell rank \((2^\kappa)^+\)), as well as having many such cardinals below it. Any supercompact cardinal is also Woodin and a limit of Woodin cardinals, and much more. Supercompact cardinals possess curious reflection properties that can explain their size - if a cardinal with some property that is witnessed by a structure of limited rank exists above a supercompact cardinal, then a cardinal with that property exists below that same cardinal. For example, the least \(n\)-huge cardinal, if it exists, is always less than the least supercompact cardinal; and if the generalized continuum hypothesis holds below the least supercompact cardinal, it holds everywhere. This is because the failure of GCH at a cardinal \(\nu\) can be witnessed within \(V_{\nu+2}\), and thus the existence of such a \(\nu\) above a supercompact implies its existence below a supercompact. The least supercompact cardinal in particular possesses a potent \(\Pi^1_1\)-reflection property: the least supercompact is precisely the least \(\kappa\) so that, for every structure \(\mathcal{M}\) whose domain has cardinality at least \(\kappa\) and for every \(\Pi^1_1\)-sentence it satisfies, there is some \(\mathcal{N}\) which also satisfies that sentence so that the cardinality of the domain of \(\mathcal{N}\) is less than the cardinality of the domain of \(\mathcal{N}\), and whose relations are the restrictions of the relations of \(\mathcal{M}\) to the domain of \(N\). For example, if \(\varphi\) is a formula, \(R \subseteq V_\kappa\) and \((V_\kappa, \in, R) \models \varphi\) then there is some \(M\) so that \(|M| < \kappa\) and \((M, \in, R \cap M) \models \varphi\) - this already implies inaccessibility. 36025123ef12a7fa6b6db27e2ef354f906b2872e 0 208 530 2023-09-08T18:30:56Z RhubarbJayde 25 Created page with "Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are typically either constructed - where they typically have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) and their fine structure analysed - or defined..." wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are typically either constructed - where they typically have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) and their fine structure analysed - or defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). For example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\). Also, "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. 2d7c44ad6b48cbd24b5296114249f8182e4e7953 537 530 2023-09-09T13:42:05Z RhubarbJayde 25 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are typically either constructed - where they typically have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) and their fine structure analysed - or defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. For example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\). Also, "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. 029e669c7c294acf5cf3a012e381aea66948e32a 542 537 2023-09-09T14:04:53Z RhubarbJayde 25 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are typically either constructed - where they typically have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) and their fine structure analysed - or defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. For example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\). Also, "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). fd5b6905ab556452b6f705874336d4961ea3a94f 553 542 2023-09-09T15:15:04Z RhubarbJayde 25 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). 4715f59d1a7ee585c2bd4e34370914a871d64b89 558 553 2023-09-09T15:51:01Z RhubarbJayde 25 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]]<nowiki> cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is hereditarily ordinal-definable. In fact, an even stronger theorem holds: assume the </nowiki>[[HOD dichotomy|HOD hypothesis]] holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable. ba02826423ea93bfb18fd33b544a1ea84d1ec185 0 1 531 131 2023-09-09T11:53:12Z Augigogigi 2 fixed CA link wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> e2b43ff20f800c007ae08aa90b03c1ad7f9c7348 0 209 532 2023-09-09T12:49:43Z RhubarbJayde 25 Created page with "Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible..." wikitext text/x-wiki Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \j: (V_{\lambda+1} \to V_{\pi(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). b5cee03b15203c4758204d32e9371b3952432ecc 533 532 2023-09-09T12:57:23Z RhubarbJayde 25 wikitext text/x-wiki Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \(j: (V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). 730e8b275b9b75c55887d6d66b3bf3f0e25b79e2 534 533 2023-09-09T13:32:14Z RhubarbJayde 25 wikitext text/x-wiki Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \(j: (V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). While the latter clearly characterization connects extendibles to supercompacts, the same can not be said for the first. However, this follows from Magidor's lemma: a cardinal \(\kappa\) is supercompact iff, for all \(\lambda > \kappa\), there exist \(\bar{\kappa} < \bar{\lambda} < \kappa\) and an elementary embedding \(j: V_{\bar{\lambda}+1} \to V_{\lambda+1}\) with critical point \(\bar{\kappa}\) so that \(j(\bar{\kappa}) = \kappa\). 0358deb5f3cb504318e716124f9c1a02d1f459db 535 534 2023-09-09T13:37:09Z RhubarbJayde 25 wikitext text/x-wiki Extendible cardinals are a powerful [[large cardinal]] notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \(j: (V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). While the latter clearly characterization connects extendibles to supercompacts, the same can not be said for the first. However, this follows from Magidor's lemma: a cardinal \(\kappa\) is supercompact iff, for all \(\lambda > \kappa\), there exist \(\bar{\kappa} < \bar{\lambda} < \kappa\) and an elementary embedding \(j: V_{\bar{\lambda}+1} \to V_{\lambda+1}\) with critical point \(\bar{\kappa}\) so that \(j(\bar{\kappa}) = \kappa\). If \(N\) is a [[Extender model|weak extender model]]<nowiki> for \(\kappa\)'s supercompactness, then, for all \(a \in V_\lambda\), the above characterisation of supercompactness holds and, for some \(\bar{a} \in V_{\bar{\lambda}}\):</nowiki> * \(j(\bar{a}) = a\). * <nowiki>\(j(N \cap V_{\bar{\lambda}}) = N \cap V_{\bar{\lambda}}\).</nowiki> * <nowiki>\(j \upharpoonright (N \cap V_{\bar{\lambda}}) \in N\).</nowiki> 3d06eadf200b18771b3189392bf74b445b7fe2c3 555 535 2023-09-09T15:17:40Z RhubarbJayde 25 wikitext text/x-wiki Extendible cardinals are a powerful [[large cardinal]] notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \(j: V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). While the latter clearly characterization connects extendibles to supercompacts, the same can not be said for the first. However, this follows from Magidor's lemma: a cardinal \(\kappa\) is supercompact iff, for all \(\lambda > \kappa\), there exist \(\bar{\kappa} < \bar{\lambda} < \kappa\) and an elementary embedding \(j: V_{\bar{\lambda}+1} \to V_{\lambda+1}\) with critical point \(\bar{\kappa}\) so that \(j(\bar{\kappa}) = \kappa\). If \(N\) is a [[Extender model|weak extender model]]<nowiki> for \(\kappa\)'s supercompactness, then, for all \(a \in V_\lambda\), the above characterisation of supercompactness holds and, for some \(\bar{a} \in V_{\bar{\lambda}}\):</nowiki> * \(j(\bar{a}) = a\). * <nowiki>\(j(N \cap V_{\bar{\lambda}}) = N \cap V_{\bar{\lambda}}\).</nowiki> * <nowiki>\(j \upharpoonright (N \cap V_{\bar{\lambda}}) \in N\).</nowiki> e6d689183376f08b2f6a52b86421fefbdf44e82c 0 210 536 2023-09-09T13:37:56Z RhubarbJayde 25 Created page with "The covering property is a property of [[Inner model theory|inner models]], in particular core models, which is a measure of how "closely they approximate" the real universe \(V\) of sets. Namely, we say an inner model \(N\) has the covering property iff it is able to "cover" uncountable sets of ordinals: for every uncountable set \(X\) of ordinals, there is \(Y \in N\) so that \(X \subset Y\) and \(|X| = |Y|\). Of course, \(V\) has the covering property, and so \(V = L\..." wikitext text/x-wiki The covering property is a property of [[Inner model theory|inner models]], in particular core models, which is a measure of how "closely they approximate" the real universe \(V\) of sets. Namely, we say an inner model \(N\) has the covering property iff it is able to "cover" uncountable sets of ordinals: for every uncountable set \(X\) of ordinals, there is \(Y \in N\) so that \(X \subset Y\) and \(|X| = |Y|\). Of course, \(V\) has the covering property, and so \(V = L\) implies all inner models have the covering property. It is known that the existence of [[Zero sharp|\(0^\sharp\)]] is equivalent to \(L\) not having the covering property and, in general, the existence of the [[sharp]] for \(N\) is equivalent to \(N\) not having the covering property. The covering property is intrinsically related to cardinal-related "closeness" to the universe. Namely, \(N\) has the covering property iff there is some singular cardinal \(\lambda\) which is singular in \(L\) or there is some successor cardinal \(\lambda\) which is a limit cardinal in \(L\); iff there is some singular cardinal \(\lambda\) so that, \((\lambda^+)^L < \lambda^+\). This hierarchy can be refined and stratified: we say \(N\) has the \(\delta\)-covering property, for a cardinal \(\delta\), if, for every \(X \subseteq N\) with \(|X| < \delta\), there is \(Y \in N\) so that \(X \subset Y\) and \(|Y| < \delta\). Every inner model has the \(\omega\)-covering property, and having the covering property is equivalent to having the \(\delta\)-covering property for all \(\delta > \omega_1\). This stratification is important and related to [[Extender model|weak extender models for supercompactness]]. Namely, if \(\delta\) is a [[supercompact]] cardinal and \(N\) is a weak extender model for \(\delta\)'s supercompactness, then: * \(N\) has the \(\delta\)-covering property * If \(\lambda > \delta\) is [[Cofinality|regular]] in \(N\), then the [[cardinality]] of \(\lambda\) is regular. * If \(\lambda > \delta\) is singular, then \(\lambda\) is singular in \(N\) and \((\lambda^+)^N = \lambda^+\). This shows a paradigm shift, in that weak extender models for supercompactness must be close to the universe, regardless of what further sharps and large cardinals exist. c76148154791dc0e3be1f9815a9a73a60037f60c 0 211 538 2023-09-09T13:52:22Z RhubarbJayde 25 Created page with "Kunen's inconsistency is a theorem proved by Kenneth Kunen which proves that certain [[Large cardinal|large cardinals]], which were previously believed to be natural generalizations of weaker large cardinals, are inconsistent. The proof uses the [[axiom of choice]] in a crucial way - it is believed that the large cardinals the theorem rules out may still be able to exist in [[ZFC|ZF]]. It originally proved: "there is no nontrivial elementary embedding \(j: V \to V\), the..." wikitext text/x-wiki Kunen's inconsistency is a theorem proved by Kenneth Kunen which proves that certain [[Large cardinal|large cardinals]], which were previously believed to be natural generalizations of weaker large cardinals, are inconsistent. The proof uses the [[axiom of choice]] in a crucial way - it is believed that the large cardinals the theorem rules out may still be able to exist in [[ZFC|ZF]]. It originally proved: "there is no nontrivial elementary embedding \(j: V \to V\), therefore there do not exist Reinhardt cardinals". However, its proof can also be generalized to show: * There are no \(\omega\)-huge cardinals. That is, there is no inner model \(M\) and nontrivial elementary embedding \(j: V \to M\) so that \(M^\lambda \subseteq M\), where \(\lambda = \sup\{j^n(\kappa): n < \omega\}\). * For all limit cardinals \(\lambda\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\). * If \(\kappa\) is [[supercompact]], \(N\) is a [[Extender model|weak extender model]] for \(\kappa\)'s supercompactness, and \(j: N \to N\) is a nontrivial elementary embedding, then the critical point of \(j\) is less than \(\kappa\). However, it is believed that it can not be generalized to show the nonexistence of a nontrivial elementary embedding \(j: V_{\lambda+1} \to V_{\lambda+1}\). Therefore, the rank-into-rank cardinals are believed to be inconsistent, but barely teetering on the brink of inconsistency. c182b43afdbedbffdaf61897f6220333f7acd7a6 543 538 2023-09-09T14:08:24Z RhubarbJayde 25 wikitext text/x-wiki Kunen's inconsistency is a theorem proved by Kenneth Kunen which proves that certain [[Large cardinal|large cardinals]], which were previously believed to be natural generalizations of weaker large cardinals, are inconsistent. The proof uses the [[axiom of choice]] in a crucial way - it is believed that the large cardinals the theorem rules out may still be able to exist in [[ZFC|ZF]]. It originally proved: "there is no nontrivial elementary embedding \(j: V \to V\), therefore there do not exist Reinhardt cardinals". However, its proof can also be generalized to show: * There are no \(\omega\)-huge cardinals. That is, there is no inner model \(M\) and nontrivial elementary embedding \(j: V \to M\) so that \(M^\lambda \subseteq M\), where \(\lambda = \sup\{j^n(\kappa): n < \omega\}\). * For all limit cardinals \(\lambda\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\). However, it is believed that it can not be generalized to show the nonexistence of a nontrivial elementary embedding \(j: V_{\lambda+1} \to V_{\lambda+1}\). Therefore, the rank-into-rank cardinals are believed to be inconsistent, but barely teetering on the brink of inconsistency. Kunen's inconsistency also implies that if \(\kappa\) is [[supercompact]], \(N\) is a [[Extender model|weak extender model]] for \(\kappa\)'s supercompactness, and \(j: N \to N\) is a nontrivial elementary embedding, then the critical point of \(j\) is less than \(\kappa\). It is known that there must be such an elementary embedding. f292acf7d5a3940922d669cb1f5b3adc609bc4e8 564 543 2023-09-10T21:26:13Z C7X 9 (Kunen's proof may not be generalizable to j : V_(λ+1) -> V_(λ+1), but what if something else is?) wikitext text/x-wiki Kunen's inconsistency is a theorem proved by Kenneth Kunen which proves that certain [[Large cardinal|large cardinals]], which were previously believed to be natural generalizations of weaker large cardinals, are inconsistent. The proof uses the [[axiom of choice]] in a crucial way - it is believed that the large cardinals the theorem rules out may still be able to exist in [[ZFC|ZF]]. It originally proved: "there is no nontrivial elementary embedding \(j: V \to V\), therefore there do not exist Reinhardt cardinals". However, its proof can also be generalized to show: * There are no \(\omega\)-huge cardinals. That is, there is no inner model \(M\) and nontrivial elementary embedding \(j: V \to M\) so that \(M^\lambda \subseteq M\), where \(\lambda = \sup\{j^n(\kappa): n < \omega\}\). * For all limit cardinals \(\lambda\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\). However, it is believed that it can not be generalized to show the nonexistence of a nontrivial elementary embedding \(j: V_{\lambda+1} \to V_{\lambda+1}\). Therefore, the rank-into-rank cardinals are believed to be consistent, but barely teetering on the brink of inconsistency. Kunen's inconsistency also implies that if \(\kappa\) is [[supercompact]], \(N\) is a [[Extender model|weak extender model]] for \(\kappa\)'s supercompactness, and \(j: N \to N\) is a nontrivial elementary embedding, then the critical point of \(j\) is less than \(\kappa\). It is known that there must be such an elementary embedding. 996a6ac2f76fef38216b8923d7013b12b32a17fe 0 179 539 451 2023-09-09T13:53:32Z RhubarbJayde 25 wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]], but not \(\Sigma^2_1\)-indescribable. This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are compact.<ref>https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large. 810bdfa6fa8bc30181325185d9dd25409393eea3 0 212 540 2023-09-09T13:59:24Z RhubarbJayde 25 Created page with "An extender is a collection of [[Filter|ultrafilters]] which, when combined, are able to coherently code a single elementary embedding (this can be either a nontrivial elementary embedding between a universe and an inner model, or a cofinal elementary embedding between two models of [[ZFC]] minus the powerset axiom). Namely, an extender consists of ultrafilters \(E_a\), where \(a\) is a finite set of ordinals, which cohere in a certain way, so that one is able to take th..." wikitext text/x-wiki An extender is a collection of [[Filter|ultrafilters]] which, when combined, are able to coherently code a single elementary embedding (this can be either a nontrivial elementary embedding between a universe and an inner model, or a cofinal elementary embedding between two models of [[ZFC]] minus the powerset axiom). Namely, an extender consists of ultrafilters \(E_a\), where \(a\) is a finite set of ordinals, which cohere in a certain way, so that one is able to take the direct limit of the ultrapowers of the universe. Extenders, or sequences of them, are used to build [[Inner model theory|inner models]] (more precisely, [[Extender model|extender models]]), which possess the fine structure of [[Constructible hierarchy|\(L\)]] yet witness the existence of elementary embeddings of the universe - namely, one starts with \(L\) and then "enriches" the definable powerset operator by allowing a predicate for the set \(\{(a,x): x \in E_a\}\). Therefore the model sees such elementary embeddings while still satisfying the [[axiom of choice]]. 917cfe4edf965b961363fb60a0ff3380b3163aee 541 540 2023-09-09T14:00:33Z RhubarbJayde 25 wikitext text/x-wiki An extender is a collection of [[Filter|ultrafilters]] which, when combined, are able to coherently code a single elementary embedding (this can be either a nontrivial elementary embedding between a universe and an inner model, or a cofinal elementary embedding between two models of [[ZFC]] minus the powerset axiom). Namely, an extender consists of ultrafilters \(E_a\), where \(a\) is a finite set of ordinals, which cohere in a certain way, so that one is able to take the direct limit of the ultrapowers with respect to the elements of the extender, and then the transitive collapse of that model. Extenders, or sequences of them, are used to build [[Inner model theory|inner models]] (more precisely, [[Extender model|extender models]]), which possess the fine structure of [[Constructible hierarchy|\(L\)]] yet witness the existence of elementary embeddings of the universe - namely, one starts with \(L\) and then "enriches" the definable powerset operator by allowing a predicate for the set \(\{(a,x): x \in E_a\}\). Therefore the model sees such elementary embeddings and so accommodates measurable, strong, etc. cardinals while still satisfying the [[axiom of choice]]. 22911aaa33a11a7a48cb090a861d6521cae8b0a4 0 143 544 422 2023-09-09T14:10:46Z RhubarbJayde 25 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (\(L\) does not have the [[covering property]]).<ref>Any text about Jensen's covering theorem</ref> * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref> * There is a proper class of nontrivial elementary embeddings \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\).<ref name=":0">W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref> * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\).<ref name=":0" /> While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. Alternatively, \(0^\sharp\) may be defined as a sound mouse (iterable premouse), or as an Ehrenfeucht-Mostowski blueprint. 4012e859883a889a2d0b080a25f079ba727a42ce 1 169 545 412 2023-09-09T14:12:32Z RhubarbJayde 25 wikitext text/x-wiki == "Totally stable" == Does "totally stable" in "every uncountable cardinal is totally stable" mean \(L_\kappa\prec L\) for every uncountable cardinal \(\kappa\), or \(L_\kappa\prec_{\Sigma_1}L\)? I think the former is proven in Barwise (I'd have to check), and since using "totally stable" to refer to \(L_\kappa\prec_{\Sigma_1}L\) is pretty much only confined to googology I'm not sure [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 22:57, 31 August 2023 (UTC) : It means \(L_\kappa \prec L\) for every uncountable cardinal \(\kappa\). Interestingly, the case for all \(\kappa\) follows from a single case: \(0^\sharp\)'s existence is equivalent to \(L_{\aleph_\omega} \prec L\). [[User:RhubarbJayde|RhubarbJayde]] ([[User talk:RhubarbJayde|talk]]) 14:12, 9 September 2023 (UTC) c1f9348b7e000d177eabe6283cc2aa4c067e6611 0 213 546 2023-09-09T14:47:03Z RhubarbJayde 25 Created page with "The HOD dichotomy theorem is a theorem which shows that HOD, the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\)...." wikitext text/x-wiki The HOD dichotomy theorem is a theorem which shows that HOD, the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two equivalents. Similarly, the HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). * <nowiki>For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).</nowiki> However, there is no known [[sharp]] for HOD that would cause the first option to hold. An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is [[Countability|uncountable]] [[Cofinality|regular]]<nowiki> and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an </nowiki>[[extendible]] cardinal, then the following are equivalent: * \(\mathrm{HOD}\) is a [[Extender model|weak extender model]] for \(\delta\)'s supercompactness. * There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. Then we have a strong dichotomy: if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. * No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. Neither of these are particularly hard to prove. 5fea1c49a288ad0fdb65acfa69949735534842c0 547 546 2023-09-09T14:47:22Z RhubarbJayde 25 wikitext text/x-wiki The HOD dichotomy theorem is a theorem which shows that [[Ordinal definable|HOD]], the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two equivalents. Similarly, the HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). * <nowiki>For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).</nowiki> However, there is no known [[sharp]] for HOD that would cause the first option to hold. An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is [[Countability|uncountable]] [[Cofinality|regular]]<nowiki> and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an </nowiki>[[extendible]] cardinal, then the following are equivalent: * \(\mathrm{HOD}\) is a [[Extender model|weak extender model]] for \(\delta\)'s supercompactness. * There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. Then we have a strong dichotomy: if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. * No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. Neither of these are particularly hard to prove. 5e2ffa6c0409d8862b0eacf9792e91a64fd31c50 551 547 2023-09-09T15:11:39Z RhubarbJayde 25 wikitext text/x-wiki The HOD dichotomy theorem is a theorem which shows that [[Ordinal definable|HOD]], the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two useful equivalents. Similarly, the (weak) HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). * <nowiki>For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).</nowiki> However, there is no known [[sharp]] for HOD that would cause the first option to hold. An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is [[Countability|uncountable]] [[Cofinality|regular]]<nowiki> and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an </nowiki>[[extendible]] cardinal, then the following are equivalent: * \(\mathrm{HOD}\) is a [[Extender model|weak extender model]] for \(\delta\)'s supercompactness. * There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. Then we have a strong dichotomy: if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. * No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. Neither of these are particularly hard to prove. 1357c796e264c36e6652850c22f0d6206646ada9 552 551 2023-09-09T15:12:23Z RhubarbJayde 25 wikitext text/x-wiki The HOD dichotomy theorem is a theorem which shows that [[Ordinal definable|HOD]], the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two useful equivalents. Similarly, the (weak) HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). * <nowiki>For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).</nowiki> However, there is no known [[sharp]] for HOD that would cause the first option to hold. An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is [[Countability|uncountable]] [[Cofinality|regular]]<nowiki> and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an </nowiki>[[extendible]] cardinal, then the following are equivalent: * \(\mathrm{HOD}\) is a [[Extender model|weak extender model]] for \(\delta\)'s supercompactness. * There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. Then we have a strong dichotomy: if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. * No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. Neither of these three statements are particularly hard to prove. 9db30036ae8862c4eda9ab3f8eb8727f17c2b5b7 557 552 2023-09-09T15:23:35Z RhubarbJayde 25 wikitext text/x-wiki The HOD dichotomy theorem is a theorem which shows that [[Ordinal definable|HOD]], the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two useful equivalents. Similarly, the (weak) HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). * <nowiki>For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).</nowiki> However, there is no known [[sharp]] for HOD that would cause the first option to hold. An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is [[Countability|uncountable]] [[Cofinality|regular]]<nowiki> and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an </nowiki>[[extendible]] cardinal, then the following are equivalent: * \(\mathrm{HOD}\) is a [[Extender model|weak extender model]] for \(\delta\)'s supercompactness. * There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. Then we have a strong dichotomy: if \(\delta\) is an extendible cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. * No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. None of these three statements are particularly hard to prove. The HOD hypothesis says that there is a proper class of cardinals \(\lambda\) which are not \(\omega\)-strongly measurable in HOD: therefore, if there is an extendible cardinal and the HOD hypothesis holds, then \(\mathrm{HOD}\) is close to \(V\). 4adbf53596994b471db77bdfcd7a998e1dd88202 559 557 2023-09-09T16:07:53Z RhubarbJayde 25 wikitext text/x-wiki The HOD dichotomy theorem is a theorem which shows that [[Ordinal definable|HOD]], the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in [[Constructible hierarchy|\(L\)]]. * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two useful equivalents. Similarly, the (weak) HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). * <nowiki>For every singular \(\gamma > \delta\), \(\gamma\) is singular in \(\mathrm{HOD}\) and \((\gamma^+)^{\mathrm{HOD}} = \gamma^+\).</nowiki> However, there is no known [[sharp]] for HOD that would cause the first option to hold. An even stronger theorem holds: say \(\lambda\) is \(\omega\)-strongly measurable in HOD iff it is [[Countability|uncountable]] [[Cofinality|regular]]<nowiki> and there is \(\kappa < \lambda\) so that \((2^\kappa)^{\mathrm{HOD}} < \lambda\) and, for any partition of \(\{\alpha < \lambda: \mathrm{cof}(\alpha) = \omega\}\) into stationary sets \(\langle S_\alpha: \alpha < \kappa \rangle\), we have \(\langle S_\alpha: \alpha < \kappa \rangle \notin \mathrm{HOD}\). This definition is inspired by Solovay's splitting theorem. Any cardinal which is \(\omega\)-strongly measurable in HOD is measurable in \(\mathrm{HOD}\), and in fact, we have an interesting equivalence: if \(\delta\) is an </nowiki>[[extendible]] cardinal, then the following are equivalent: * \(\mathrm{HOD}\) is a [[Extender model|weak extender model]] for \(\delta\)'s supercompactness. * There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. Then we have a strong dichotomy: if \(\delta\) is an extendible cardinal, either: * Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. * No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. None of these three statements are particularly hard to prove. The HOD hypothesis says that there is a proper class of cardinals \(\lambda\) which are not \(\omega\)-strongly measurable in HOD: therefore, if there is an extendible cardinal and the HOD hypothesis holds, then \(\mathrm{HOD}\) is close to \(V\). Currently, it's not known if the successor of a singular strong limit cardinal of uncountable cofinality can ever be \(\omega\)-strongly measurable in HOD - therefore, there is reason to believe in the HOD hypothesis. Interestingly, the HOD hypothesis implies the following. Letting \(T = \mathrm{Th}_{\Sigma_2}(V)\) be the \(\Sigma_2\)-theory of \(V\) with ordinal parameters (note that the notation \(\mathrm{Th}_{\Sigma_2}(V)\) typically indicates allowance arbitrary set parameters), i.e. \(T\) contains all the true formulae which are [[ZFC|ZF]]-provably equivalent to a formula of the form \(\exists \alpha (V_\alpha \models \psi(\beta_1, \beta_2, \cdots, \beta_n)\) for an arbitrary formula \(\psi\) and ordinals \(\beta_1, \beta_2, \cdots, \beta_n\). Then there is no nontrivial elementary embedding \(j: (\mathrm{HOD}, T) \to (\mathrm{HOD}, T)\). The HOD conjecture is the assertion that the theory of ZFC plus "there is a [[Supercompact|supercompact cardinal]]" proves the HOD hypothesis. Unlike the HOD hypothesis itself, this is an arithmetic statement. The HOD conjecture has some remarkable consequences, which don't require the [[axiom of choice]]: one of the most surprising is a possibility to eliminate the usage of choice within the proof of [[Kunen's inconsistency]]. Namely, if the HOD conjecture is true and \(\delta\) is extendible, then, for all \(\lambda > \delta\), there is no nontrivial elementary embedding \(j: V_{\lambda+2} \to V_{\lambda+2}\), and this doesn't require the axiom of choice. 644668765692eeeade960aa8abc9c22d956ad54e 0 214 548 2023-09-09T15:06:10Z RhubarbJayde 25 Created page with "Ordinal definability is a concept which is key in certain aspects of [[inner model theory]]. We say a set \(x\) is ordinal-definable iff there is an ordinal \(\beta\) and a collection \(\alpha_1, \alpha_2, \cdots, \alpha_n\) of ordinals so that \(x \in V_\beta\) and there is a first-order formula \(\varphi\) such that, for all \(y \in V_\beta\), \(y = x\) iff \(V_\beta \models \varphi(y, \alpha_1, \alpha_2, \cdots, \alpha_n)\). In other words, it is definable at some ini..." wikitext text/x-wiki Ordinal definability is a concept which is key in certain aspects of [[inner model theory]]. We say a set \(x\) is ordinal-definable iff there is an ordinal \(\beta\) and a collection \(\alpha_1, \alpha_2, \cdots, \alpha_n\) of ordinals so that \(x \in V_\beta\) and there is a first-order formula \(\varphi\) such that, for all \(y \in V_\beta\), \(y = x\) iff \(V_\beta \models \varphi(y, \alpha_1, \alpha_2, \cdots, \alpha_n)\). In other words, it is definable at some initial segment of the von Neumann hierarchy with ordinal parameters. Unlike with other inner models, HOD is not particularly robust: for any model \(M\), there is a forcing extension \(M[G]\) in which all elements of \(M\) become ordinal definable (i.e. \(V = \mathrm{HOD}\)) - in contrast with the fact that e.g. being constructible can not be changed by forcing. On the other hand, HOD is able to accommodate many large cardinals. This is in contrast to the fact that traditional inner models that can accommodate even just [[Supercompact|supercompact cardinals]] have not yet been constructed. For example, "there is a [[measurable]] cardinal and all sets are constructible, i.e. \(V = L\)" is necessarily inconsistent, while the consistency of an [[Extendible|extendible cardinal]] implies the consistency of "there is a measurable cardinal and all sets are ordinal definable, i.e. \(V = \mathrm{HOD}\)". This follows from the [[HOD dichotomy]]. On the more extreme side of things, if the wholeness axioms, an axiomatization of the existence of a nontrivial elementary embedding of the universe into itself which avoid [[Kunen's inconsistency]], are consistent, then they are compatible with the assertion "all sets are ordinal definable". HOD is a useful inner model and the aforementioned HOD dichotomy gives an analogous result for HOD to the fact that [[Zero sharp|\(0^\sharp\)]] exists iff L does not have the [[covering property]]: however, there is no known [[sharp]] for HOD that would lead to HOD being far away from the true universe, primarily due to the fact that it is compatible with all known large cardinals. The reason traditional inner models are preferred is that, as mentioned earlier, not only is HOD non-robust, but it does not have the same fine structure that [[Constructible hierarchy|\(L\)]] and core models have. 9435371bde42f76b259300b24d125314b529d28b 549 548 2023-09-09T15:08:08Z RhubarbJayde 25 wikitext text/x-wiki Ordinal definability is a concept which is key in certain aspects of [[inner model theory]]. We say a set \(x\) is ordinal-definable iff there is an ordinal \(\beta\) and a collection \(\alpha_1, \alpha_2, \cdots, \alpha_n\) of ordinals so that \(x \in V_\beta\) and there is a first-order formula \(\varphi\) such that, for all \(y \in V_\beta\), \(y = x\) iff \(V_\beta \models \varphi(y, \alpha_1, \alpha_2, \cdots, \alpha_n)\). In other words, it is definable at some initial segment of the von Neumann hierarchy with ordinal parameters. The class of ordinal definable sets is denoted \(\mathrm{OD}\), and \(\mathrm{HOD} = \{x: \mathrm{TC}(x) \in \mathrm{OD}\}\). \(\mathrm{HOD}\) is an inner model and \(V = \mathrm{HOD}\) is equivalent to \(V = \mathrm{OD}\): however, \(\mathrm{OD}\) is rarely talked about because, in the case that \(V \neq \mathrm{OD}\), \(\mathrm{OD}\) may not be transitive, while \(\mathrm{HOD}\) will, trivially, always be. \(\mathrm{HOD}\) is one of the simplest and well-known [[Inner model theory|inner models]]. Unlike with other inner models, HOD is not particularly robust: for any model \(M\), there is a forcing extension \(M[G]\) in which all elements of \(M\) become ordinal definable (i.e. \(V = \mathrm{HOD}\)) - in contrast with the fact that e.g. being constructible can not be changed by forcing. On the other hand, HOD is able to accommodate many large cardinals. This is in contrast to the fact that traditional inner models that can accommodate even just [[Supercompact|supercompact cardinals]] have not yet been constructed. For example, "there is a [[measurable]] cardinal and all sets are constructible, i.e. \(V = L\)" is necessarily inconsistent, while the consistency of an [[Extendible|extendible cardinal]] implies the consistency of "there is a measurable cardinal and all sets are ordinal definable, i.e. \(V = \mathrm{HOD}\)". This follows from the [[HOD dichotomy]]. On the more extreme side of things, if the wholeness axioms, an axiomatization of the existence of a nontrivial elementary embedding of the universe into itself which avoid [[Kunen's inconsistency]], are consistent, then they are compatible with the assertion "all sets are ordinal definable". HOD is a useful inner model and the aforementioned HOD dichotomy gives an analogous result for HOD to the fact that [[Zero sharp|\(0^\sharp\)]] exists iff L does not have the [[covering property]]: however, there is no known [[sharp]] for HOD that would lead to HOD being far away from the true universe, primarily due to the fact that it is compatible with all known large cardinals. The reason traditional inner models are preferred is that, as mentioned earlier, not only is HOD non-robust, but it does not have the same fine structure that [[Constructible hierarchy|\(L\)]] and core models have. c94935d86cbbe5c4b35b66d89b71ae4d6ea22941 550 549 2023-09-09T15:10:34Z RhubarbJayde 25 wikitext text/x-wiki Ordinal definability is a concept which is key in certain aspects of [[inner model theory]]. We say a set \(x\) is ordinal-definable iff there is an ordinal \(\beta\) and a collection \(\alpha_1, \alpha_2, \cdots, \alpha_n\) of ordinals so that \(x \in V_\beta\) and there is a first-order formula \(\varphi\) such that, for all \(y \in V_\beta\), \(y = x\) iff \(V_\beta \models \varphi(y, \alpha_1, \alpha_2, \cdots, \alpha_n)\). In other words, it is definable at some initial segment of the von Neumann hierarchy with ordinal parameters. The class of ordinal definable sets is denoted \(\mathrm{OD}\), and \(\mathrm{HOD} = \{x: \mathrm{TC}(x) \in \mathrm{OD}\}\), where \(\mathrm{TC}\) denotes transitive closure. \(\mathrm{HOD}\) is an inner model and \(V = \mathrm{HOD}\) is equivalent to \(V = \mathrm{OD}\): however, \(\mathrm{OD}\) is rarely talked about because, in the case that \(V \neq \mathrm{OD}\), \(\mathrm{OD}\) may not be transitive, while \(\mathrm{HOD}\) will, trivially, always be. \(\mathrm{HOD}\) is one of the simplest and well-known [[Inner model theory|inner models]]. Unlike with other inner models, HOD is not particularly robust: for any model \(M\), there is a forcing extension \(M[G]\) in which all elements of \(M\) become ordinal definable (i.e. \(V = \mathrm{HOD}\)) - in contrast with the fact that e.g. being constructible can not be changed by forcing. On the other hand, HOD is able to accommodate many large cardinals. This is in contrast to the fact that traditional inner models that can accommodate even just [[Supercompact|supercompact cardinals]] have not yet been constructed. For example, "there is a [[measurable]] cardinal and all sets are constructible, i.e. \(V = L\)" is necessarily inconsistent, while the consistency of an [[Extendible|extendible cardinal]] implies the consistency of "there is a measurable cardinal and all sets are ordinal definable, i.e. \(V = \mathrm{HOD}\)". This follows from the [[HOD dichotomy]]. On the more extreme side of things, if the wholeness axioms, an axiomatization of the existence of a nontrivial elementary embedding of the universe into itself which avoid [[Kunen's inconsistency]], are consistent, then they are compatible with the assertion "all sets are ordinal definable". HOD is a useful inner model and the aforementioned HOD dichotomy gives an analogous result for HOD to the fact that [[Zero sharp|\(0^\sharp\)]] exists iff L does not have the [[covering property]]: however, there is no known [[sharp]] for HOD that would lead to HOD being far away from the true universe, primarily due to the fact that it is compatible with all known large cardinals. The reason traditional inner models are preferred is that, as mentioned earlier, not only is HOD non-robust, but it does not have the same fine structure that [[Constructible hierarchy|\(L\)]] and core models have. ea9ca1c5df607ae914e277cbb19a57f643046597 556 550 2023-09-09T15:20:05Z RhubarbJayde 25 wikitext text/x-wiki Ordinal definability is a concept which is key in certain aspects of [[inner model theory]]. We say a set \(x\) is ordinal-definable iff there is an ordinal \(\beta\) and a collection \(\alpha_1, \alpha_2, \cdots, \alpha_n\) of ordinals so that \(x \in V_\beta\) and there is a first-order formula \(\varphi\) such that, for all \(y \in V_\beta\), \(y = x\) iff \(V_\beta \models \varphi(y, \alpha_1, \alpha_2, \cdots, \alpha_n)\). In other words, it is definable at some initial segment of the von Neumann hierarchy with ordinal parameters. The class of ordinal definable sets is denoted \(\mathrm{OD}\), and \(\mathrm{HOD} = \{x: \mathrm{TC}(x) \in \mathrm{OD}\}\), where \(\mathrm{TC}\) denotes transitive closure. \(\mathrm{HOD}\) is an inner model and \(V = \mathrm{HOD}\) is equivalent to \(V = \mathrm{OD}\): however, \(\mathrm{OD}\) is rarely talked about because, in the case that \(V \neq \mathrm{OD}\), \(\mathrm{OD}\) may not be transitive, while \(\mathrm{HOD}\) will, trivially, always be. \(\mathrm{HOD}\) is one of the simplest and well-known [[Inner model theory|inner models]]. Unlike with other inner models, HOD is not particularly robust: for any model \(M\), there is a forcing extension \(M[G]\) in which all elements of \(M\) become ordinal definable (i.e. \(V = \mathrm{HOD}\)) - in contrast with the fact that e.g. being constructible can not be changed by forcing. On the other hand, HOD is able to accommodate many large cardinals. This is in contrast to the fact that traditional inner models that can accommodate even just [[Supercompact|supercompact cardinals]] have not yet been constructed. For example, "there is a [[measurable]] cardinal and all sets are constructible, i.e. \(V = L\)" is necessarily inconsistent, while the existence of an [[Extendible|extendible cardinal]] implies the consistency of "there is a measurable cardinal and all sets are ordinal definable, i.e. \(V = \mathrm{HOD}\)". This follows from the [[HOD dichotomy]], although in this case the hypothesis can be weakened from an extendible to a \(\omega\)-extendible cardinal. On the more extreme side of things, if the wholeness axioms, an axiomatization of the existence of a nontrivial elementary embedding of the universe into itself which avoid [[Kunen's inconsistency]], are consistent, then they are compatible with the assertion "all sets are ordinal definable".HOD is a useful inner model and the aforementioned HOD dichotomy gives an analogous result for HOD to the fact that [[Zero sharp|\(0^\sharp\)]] exists iff L does not have the [[covering property]]: however, there is no known [[sharp]] for HOD that would lead to HOD being far away from the true universe, primarily due to the fact that it is compatible with all known large cardinals. The reason traditional inner models are preferred is that, as mentioned earlier, not only is HOD non-robust, but it does not have the same fine structure that [[Constructible hierarchy|\(L\)]] and core models have. 43e97504f8f7a149f62bcd3a40fcceea82ca71c2 0 215 554 2023-09-09T15:16:35Z RhubarbJayde 25 Redirected page to [[Ordinal definable]] wikitext text/x-wiki #REDIRECT [[Ordinal definable]] fd924c72d9ade931c88f0fb09408e473d0e2317f 0 133 560 404 2023-09-09T16:29:36Z RhubarbJayde 25 wikitext text/x-wiki The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the [[Ordinal|ordinals]]. Therefore, it is important in [[inner model theory]], as well as in the study of \(\alpha\)-recursion theory, [[stability]], [[Gandy ordinal|Gandy ordinals]] and [[Reflection principle|reflection principles]]. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann definition of ordinal, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, there are \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) are of importance when \(Y\) is uncountable, to ensure that there are more than \(\aleph_0\) definable subsets of \(Y\), but they do not have any effect when \(Y\supseteq\mathbb N\) is countable, since all elements of the natural numbers are definable. Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref> * \(L_0\) is the empty set. * \(L_{\alpha+1}\) is the collection of definable subsets of \(L_\alpha\). * For limit \(\alpha\), \(L_\alpha\) is the union of \(L_\beta\) for \(\beta < \alpha\). Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies. This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\).<ref>Most set theory texts</ref> If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. The constructible universe is defined analogously to \(V\), by letting \(L\) be the union of \(L_\alpha\) among all ordinals \(\alpha\). For all \(\alpha\), the ordinals which are elements of \(L_\alpha\) are precisely the ordinals less than \(\alpha\). That is, \(\mathrm{Ord} \cap L_\alpha = \alpha\). We show this by induction: * \(\mathrm{Ord} \cap L_0 = \mathrm{Ord} \cap \emptyset = \mathrm{Ord} \cap 0 = 0\). * Assume \(\mathrm{Ord} \cap L_\alpha = \alpha\). Then \(\alpha\) is a definable subset of \(L_\alpha\) since \(\alpha\) is just the set of \(x \in L_\alpha\) which \(L_\alpha\) thinks are ordinals, so \(\alpha \in L_{\alpha+1}\). For \(\beta > \alpha\), \(\alpha \notin L_\alpha\) yet \(\alpha \in \beta\), therefore \(\beta\) is not a definable subset of \(L_\alpha\) and so \(\beta \notin L_{\alpha+1}\). Lastly, we have \(\gamma \in L_{\alpha+1}\) for \(\gamma < \alpha\), therefore \(\mathrm{Ord} \cap L_\alpha = \alpha \cup \{\alpha\} = \alpha+1\). * If \(\alpha\) is a limit ordinal, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap (\bigcup_{\beta < \alpha} L_\beta) = \bigcup_{\beta < \alpha} (\mathrm{Ord} \cap L_\beta) = \bigcup_{\beta < \alpha} \beta = \alpha\). In particular, \(\mathrm{Ord} \cap L_\alpha = \mathrm{Ord} \cap V_\alpha\) for all \(\alpha\), and \(L\) contains all the ordinals. A similar method can be used to show that \(L_\alpha\) is transitive for all \(\alpha\), and that \(L_\alpha \in L_\beta\) for all \(\alpha < \beta\). == Alternate characterisations == Kurt Gödel, the inventor of the constructible universe, isolated 10 basic "Gödel operations", relatively simple operations on sets which can be used to built up the universe of sets. They are: * \((X, Y) \mapsto \{X, Y\}\), i.e. the unordered pair of \(X\) and \(Y\) * \((X, Y) \mapsto X \times Y\), i.e. the Cartesian product of \(X\) and \(Y\) * \((X, Y) \mapsto \{(x,y): y \in Y \land x \in X \cap y \}\), i.e. the restriction of the elementhood relation to \(X \times Y\) * \((X, Y) \mapsto X \setminus Y\), i.e. the complement of \(Y\) relative to \(X\) * \((X, Y) \mapsto X \cap Y\), i.e. the intersection of \(X\) and \(Y\) * \(X \mapsto \bigcup X\), i.e. the union of \(X\) * \(X \mapsto \{x: \exists y ((x, y) \in X)\}\), i.e. the domain of \(X\) * \(X \mapsto \{(x,y): (y,x) \in X\}\), i.e. the inverse of \(X\) * \(X \mapsto \{(x,z,y): (x,y,z) \in X\}\) * \(X \mapsto \{(z,x,y): (x,y,z) \in X\}\) For a set \(X\), we let \(\mathrm{cl}(X)\) be the closure of \(X\) under the Gödel operations - i.e. under taking unions, intersections, permutations, and more. He proved that, for all sets \(X\), a subset of \(X\) is definable iff it is an element of \(\mathrm{cl}(X \cup \{X\})\). The proof is based on his normal form theorem, which asserts that, if \(\varphi(x_1, x_2, \cdots, x_n)\) is a \(\Delta_0\)-formula, then, for all \(X_1, X_2, \cdots, X_n\), the set \(\{(x_1, x_2, \cdots, x_n) \in X_1 \times X_2 \times \cdots \times X_n: \varphi(x_1, x_2, \cdots, x_n)\}\) is given by a composition of the Gödel operations. Therefore, we can alternatively define the constructible hierarchy like so: * \(L_0 = \emptyset\). * \(L_{\alpha+1} = \mathrm{cl}(L_\alpha \cup \{L_\alpha\}) \cap \mathcal{P}(L_\alpha)\). * For limit \(\alpha\), \(L_\alpha = \bigcup_{\beta < \alpha} L_\beta\). As we've mentioned earlier, \(L\) contains all the ordinals. And since it's closed under unions, unordered pair, and further, it is possible to show that \(L\) is a model of \(\mathrm{ZFC}\). See [https://en.wikipedia.org/w/index.php?title=Constructible_universe the Wikipedia article] for a full proof. Therefore, \(L\) is an [[Inner model theory|inner model]] (i.e. a transitive model of ZFC containing all the ordinals), and a famous theorem proves that \(L\) is the ''minimal'' inner model. That is, for all \(X \subset L\), we either have: * \(X\) does not satisfy ZFC. * There is an ordinal not in \(X\). * \(X\) is not transitive. Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\): <nowiki>Assume there is a measurable cardinal, and let \(\kappa\) be the least measurable cardinal, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki> Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself. afae078d80788085eed63992fe5f7b1d382a220b 0 208 565 558 2023-09-11T17:51:19Z 82.8.204.174 0 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]]<nowiki> cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is [[ordinal-definable]]. In fact, an even stronger theorem holds: assume the HOD hypothesis holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable. 7bee6c88ad519c86e7e421f02202bb58b3eb35d9 566 565 2023-09-11T21:50:23Z C7X 9 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is a weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]] cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is [[ordinal-definable]]. In fact, an even stronger theorem holds: assume the HOD hypothesis holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable. c5697a6487a8da4a7b5dd8f07e4f8f2eccfdd7e1 572 566 2023-09-14T16:48:30Z RhubarbJayde 25 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is the minimal weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]] cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is [[ordinal-definable]]. In fact, an even stronger theorem holds: assume the HOD hypothesis holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable. 6d06d071f44cd95abbcf051a9061f66c47ae7fb8 0 56 567 416 2023-09-13T21:25:53Z C7X 9 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>GS dimensional Veblen extensions</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> == References == <reflist /> 0878b564a5a6145d9662c4499fb172a1fd2d5050 568 567 2023-09-13T21:26:03Z C7X 9 /* References */ wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>GS dimensional Veblen extensions</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> == References == {{Reflist}} 42c476446783a06d4a4886a8a1a5cb15a7314704 569 568 2023-09-13T21:26:14Z C7X 9 /* References */ wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>GS dimensional Veblen extensions</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> == References == cdc27a20e35c65bf93377a54749279668871f5d8 603 569 2023-10-22T17:03:24Z C7X 9 wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>https://arxiv.org/abs/2310.12832v1?</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> == References == 816f4db4d5bfc96298baa7d0640a15a126d80a95 0 73 570 257 2023-09-13T21:54:11Z C7X 9 /* Longer gaps */ wikitext text/x-wiki A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).<ref name="MarekSrebrny73" />^p.368 Gap ordinals are very large. This is because, if \( \alpha \) is a gap ordinal, then \( L_\alpha \cap \mathcal{P}(\omega) \) satisfies second-order arithmetic, despite not containing ''all'' subsets of \( \omega \). Therefore, if \( \alpha \) is a gap ordinal, it is admissible, recursively inaccessible, recursively Mahlo, nonprojectible, and more. However, there can still be countable gap ordinals. There is a nice analogy between gap ordinals and cardinals. Note that \( \alpha \) is a cardinal if, for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \). If \( \alpha \) is infinite, we have \( \pi \subseteq \gamma \times \alpha \subseteq \alpha \times \alpha \subseteq V_\alpha^2 \subseteq V_\alpha \) and thus \( \pi \in V_{\alpha + 1} \). Thus, "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in V_{\alpha + 1} \)" is equivalent to being a cardinal. Meanwhile, the least ordinal satisfying the similar but weaker condition "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in L_{\alpha + 1} \)" is equal to the least gap ordinal, since it's equivalent to \( L_\alpha \) satisfying separation.<ref>R.Björn Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, Volume 4, Issue 3, 1972, Pages 229-308</ref> Harvey Friedman proved that \(L_{\beta_0}\cap\mathcal P(\omega)\) does not satisfy \(\Sigma^0_5\) determinacy.<ref>Montalbán, Shore, "[https://math.berkeley.edu/~antonio/papers/Delta4Det.pdf The Limits of Determinacy in Second Order Arithmetic]", p.22 (2011). Accessed 13 September 2023.</ref> ==Longer gaps== Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha) \cap \mathcal{P}(\omega) = \emptyset\).<ref name="MarekSrebrny73" />^p.365 If such an \( \alpha \) starts a gap, then it is said to start a gap of length \( \gamma \). It is possible for \( \alpha \) to start a gap of length \( > \alpha \): for example, the least \( \alpha \) so that \( \alpha \) starts a gap of length \( \alpha^+ \) is equal to the least admissible which is not locally countable. There can also be second-order gap, and more. An ordinal \( \alpha \) is said to start an \( \eta \)th-order gap of length \( \gamma \) if \( (L_{\alpha+\gamma} \setminus L_\alpha) \cap \mathcal{P}^\eta(\omega) = \emptyset \) and, for all \( \beta < \alpha \), \((L_\alpha \setminus L_\beta) \cap \mathcal {P}^\eta(\omega) \neq \emptyset\). The least ordinal which starts a second-order gap is greater than the least \( \alpha \) which starts a first-order gap of length \( \alpha \), and more. If \( 0^\sharp \) exists, then, for any countable \( \eta\) and any \( \gamma \) at all, there is a countable \( \delta \) which starts an \( \eta \)th-order gap of length \( \gamma \). Meanwhile, if \( V = L \), then there is no countable ordinal starting a first-order gap of length \( \omega_1 \). ==Citations== 54dd398ea8215537cd8306d3598e1e1d51878c74 571 570 2023-09-13T21:54:34Z C7X 9 wikitext text/x-wiki A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).<ref name="MarekSrebrny73" />^p.368 Gap ordinals are very large. This is because, if \( \alpha \) is a gap ordinal, then \( L_\alpha \cap \mathcal{P}(\omega) \) satisfies second-order arithmetic, despite not containing ''all'' subsets of \( \omega \). Therefore, if \( \alpha \) is a gap ordinal, it is admissible, recursively inaccessible, recursively Mahlo, nonprojectible, and more. However, there can still be countable gap ordinals. There is a nice analogy between gap ordinals and cardinals. Note that \( \alpha \) is a cardinal if, for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \). If \( \alpha \) is infinite, we have \( \pi \subseteq \gamma \times \alpha \subseteq \alpha \times \alpha \subseteq V_\alpha^2 \subseteq V_\alpha \) and thus \( \pi \in V_{\alpha + 1} \). Thus, "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in V_{\alpha + 1} \)" is equivalent to being a cardinal. Meanwhile, the least ordinal satisfying the similar but weaker condition "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in L_{\alpha + 1} \)" is equal to the least gap ordinal, since it's equivalent to \( L_\alpha \) satisfying separation.<ref>R.Björn Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, Volume 4, Issue 3, 1972, Pages 229-308</ref> Harvey Friedman proved that \(L_{\beta_0}\cap\mathcal P(\omega)\) does not satisfy \(\Sigma^0_5\) determinacy.<ref>A. Montalbán, R. Shore, "[https://math.berkeley.edu/~antonio/papers/Delta4Det.pdf The Limits of Determinacy in Second Order Arithmetic]", p.22 (2011). Accessed 13 September 2023.</ref> ==Longer gaps== Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha) \cap \mathcal{P}(\omega) = \emptyset\).<ref name="MarekSrebrny73" />^p.365 If such an \( \alpha \) starts a gap, then it is said to start a gap of length \( \gamma \). It is possible for \( \alpha \) to start a gap of length \( > \alpha \): for example, the least \( \alpha \) so that \( \alpha \) starts a gap of length \( \alpha^+ \) is equal to the least admissible which is not locally countable. There can also be second-order gap, and more. An ordinal \( \alpha \) is said to start an \( \eta \)th-order gap of length \( \gamma \) if \( (L_{\alpha+\gamma} \setminus L_\alpha) \cap \mathcal{P}^\eta(\omega) = \emptyset \) and, for all \( \beta < \alpha \), \((L_\alpha \setminus L_\beta) \cap \mathcal {P}^\eta(\omega) \neq \emptyset\). The least ordinal which starts a second-order gap is greater than the least \( \alpha \) which starts a first-order gap of length \( \alpha \), and more. If \( 0^\sharp \) exists, then, for any countable \( \eta\) and any \( \gamma \) at all, there is a countable \( \delta \) which starts an \( \eta \)th-order gap of length \( \gamma \). Meanwhile, if \( V = L \), then there is no countable ordinal starting a first-order gap of length \( \omega_1 \). ==Citations== 795e033d03fe905cf2ed6e61dd16b875c245a06a 0 216 573 2023-09-14T16:50:23Z RhubarbJayde 25 Redirected page to [[Ordinal definable]] wikitext text/x-wiki #REDIRECT [[Ordinal definable]] fd924c72d9ade931c88f0fb09408e473d0e2317f 2 217 574 2023-09-17T15:22:54Z 1ijk 28 Created page with "hi there!" wikitext text/x-wiki hi there! a903cda4b5b93d3204af0fd6b7b92d24af1923a5 0 67 575 502 2023-09-18T06:30:54Z C7X 9 For definability in V this gives something like min{α | L_α prec_Δ_1 L} wikitext text/x-wiki Countability is a key notion in set theory and apeirology. A set is called countable if it is possible to arrange its elements in a way so that they can be counted off one-by-one. In other words, it has the same [[Cardinal|size]] as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. In other words, there is a [[bijection]] from \( x \) to the natural numbers. However, this may not imply they have the same order-type, if the set is [[Well-ordered set|well-ordered]]. Georg Cantor, the founder of set theory, proved that the set of integers and the set of rational numbers are both countable, by constructing such maps. More famously, Cantor [[Cantor's diagonal argument|proved]] that the real numbers and that the powerset of the natural numbers are both uncountable, by assuming there was a map \( f \) and deriving a contradiction. In terms of ordinals, it is clear that [[Omega|\( \omega \)]]<nowiki> is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable in their rank of the constructible hierarchy. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki> Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the [[continuum hypothesis]]. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven. 49ed1531e4bfdce6f5baab9a383efb3262eeba79 0 218 576 2023-09-19T13:47:02Z C7X 9 Created page with "Uniformity is a justification used for existence of some very large large cardinals, such as measurable and strongly compact cardinals. As there is no current characterization of measurable cardinals by closure under certain operations as there are with some smaller large cardinals, uniformity is often the method of choice for justifying their existence.<ref name="MaddyI">P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]", pp.5..." wikitext text/x-wiki Uniformity is a justification used for existence of some very large large cardinals, such as measurable and strongly compact cardinals. As there is no current characterization of measurable cardinals by closure under certain operations as there are with some smaller large cardinals, uniformity is often the method of choice for justifying their existence.<ref name="MaddyI">P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]", pp.502--505. Journal of Symbolic Logic vol. 53, no. 2 (1988).</ref> The principle behind the uniformity justification states that the set-theoretic universe should remain interesting when progressing through its ranks, and that the alternative would mean "as if the universe had lost its complexity at the higher levels, as if it had flattened out, become homogeneous and boring."<ref name="MaddyI" /> Since there is a nontrivial \(aleph_0\)-additive two-valued measure on the subsets of \\(aleph_0\\), it would be expected that this property holds for larger cardinals as well, otherwise this phenomenon would occur only for \(aleph_0\) and then never reoccur for any larger cardinals. A similar justification is used for strongly compact cardinals. Given a \(mathcal L_{aleph_0,aleph_0}\)-theory \(T\), if a \(mathcal L_{aleph_0,aleph_0}\)-formula follows from \(T\), it follows from a finite subset of \\(T\\). The only infinite cardinals with this property are either \(aleph_0\) or strongly compact cardinals, and if one wants to avoid the phenomenon stopping permanently after \(aleph_0\), the existence of strongly compact cardinals must be assumed. Barwise recounts about the above:<ref>J. Bariwse, ''Admissible Sets and Structures'', p.364. Perspectives in Mathematical Logic (1975).</ref> : The remarkable argument that strongly compact cardinals exist "by analogy with \(omega\)" always reminds me of the goofang, described in ''The Book of Imaginary Beings'', by Jorge Luis Borges: :: The yarns and tall tales of the lumber camps of Wisconsin and Minnesota include some singular creatures, in which, surely, no one ever believed... :: There's another fish, the Goofang, that swims backward to keep the water out of its eyes. It's described as "about the size of a sunfish, only much bigger". 10ed38472e922502fe7f3140181dec1c3885f3c1 577 576 2023-09-19T13:47:29Z C7X 9 wikitext text/x-wiki Uniformity is a justification used for existence of some very large large cardinals, such as measurable and strongly compact cardinals. As there is no current characterization of measurable cardinals by closure under certain operations as there are with some smaller large cardinals, uniformity is often the method of choice for justifying their existence.<ref name="MaddyI">P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]", pp.502--505. Journal of Symbolic Logic vol. 53, no. 2 (1988).</ref> The principle behind the uniformity justification states that the set-theoretic universe should remain interesting when progressing through its ranks, and that the alternative would mean "as if the universe had lost its complexity at the higher levels, as if it had flattened out, become homogeneous and boring."<ref name="MaddyI" /> Since there is a nontrivial \(\aleph_0\)-additive two-valued measure on the subsets of \\(aleph_0\\), it would be expected that this property holds for larger cardinals as well, otherwise this phenomenon would occur only for \(\aleph_0\) and then never reoccur for any larger cardinals. A similar justification is used for strongly compact cardinals. Given a \(\mathcal L_{\aleph_0,\aleph_0}\)-theory \(T\), if a \(\mathcal L_{\aleph_0,\aleph_0}\)-formula follows from \(T\), it follows from a finite subset of \\(T\\). The only infinite cardinals with this property are either \(\aleph_0\) or strongly compact cardinals, and if one wants to avoid the phenomenon stopping permanently after \(\aleph_0\), the existence of strongly compact cardinals must be assumed. Barwise recounts about the above:<ref>J. Bariwse, ''Admissible Sets and Structures'', p.364. Perspectives in Mathematical Logic (1975).</ref> : The remarkable argument that strongly compact cardinals exist "by analogy with \(omega\)" always reminds me of the goofang, described in ''The Book of Imaginary Beings'', by Jorge Luis Borges: :: The yarns and tall tales of the lumber camps of Wisconsin and Minnesota include some singular creatures, in which, surely, no one ever believed... :: There's another fish, the Goofang, that swims backward to keep the water out of its eyes. It's described as "about the size of a sunfish, only much bigger". ==Citations== f91991910ed7103cca5f90d08d03352a476d8173 578 577 2023-09-19T13:47:42Z C7X 9 wikitext text/x-wiki Uniformity is a justification used for existence of some very large large cardinals, such as measurable and strongly compact cardinals. As there is no current characterization of measurable cardinals by closure under certain operations as there are with some smaller large cardinals, uniformity is often the method of choice for justifying their existence.<ref name="MaddyI">P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]", pp.502--505. Journal of Symbolic Logic vol. 53, no. 2 (1988).</ref> The principle behind the uniformity justification states that the set-theoretic universe should remain interesting when progressing through its ranks, and that the alternative would mean "as if the universe had lost its complexity at the higher levels, as if it had flattened out, become homogeneous and boring."<ref name="MaddyI" /> Since there is a nontrivial \(\aleph_0\)-additive two-valued measure on the subsets of \\(aleph_0\\), it would be expected that this property holds for larger cardinals as well, otherwise this phenomenon would occur only for \(\aleph_0\) and then never reoccur for any larger cardinals. A similar justification is used for strongly compact cardinals. Given a \(\mathcal L_{\aleph_0,\aleph_0}\)-theory \(T\), if a \(\mathcal L_{\aleph_0,\aleph_0}\)-formula follows from \(T\), it follows from a finite subset of \\(T\\). The only infinite cardinals with this property are either \(\aleph_0\) or strongly compact cardinals, and if one wants to avoid the phenomenon stopping permanently after \(\aleph_0\), the existence of strongly compact cardinals must be assumed. Barwise recounts about the above:<ref>J. Bariwse, ''Admissible Sets and Structures'', p.364. Perspectives in Mathematical Logic (1975).</ref> : The remarkable argument that strongly compact cardinals exist "by analogy with \(\omega\)" always reminds me of the goofang, described in ''The Book of Imaginary Beings'', by Jorge Luis Borges: :: The yarns and tall tales of the lumber camps of Wisconsin and Minnesota include some singular creatures, in which, surely, no one ever believed... :: There's another fish, the Goofang, that swims backward to keep the water out of its eyes. It's described as "about the size of a sunfish, only much bigger". ==References== 842952e93085618ce21a0bad907ad6d34e980287 579 578 2023-09-19T13:48:04Z C7X 9 wikitext text/x-wiki Uniformity is a justification used for existence of some very large large cardinals, such as measurable and strongly compact cardinals. As there is no current characterization of measurable cardinals by closure under certain operations as there are with some smaller large cardinals, uniformity is often the method of choice for justifying their existence.<ref name="MaddyI">P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]", pp.502--505. Journal of Symbolic Logic vol. 53, no. 2 (1988).</ref> The principle behind the uniformity justification states that the set-theoretic universe should remain interesting when progressing through its ranks, and that the alternative would mean "as if the universe had lost its complexity at the higher levels, as if it had flattened out, become homogeneous and boring."<ref name="MaddyI" /> Since there is a nontrivial \(\aleph_0\)-additive two-valued measure on the subsets of \\(aleph_0\\), it would be expected that this property holds for larger cardinals as well, otherwise this phenomenon would occur only for \(\aleph_0\) and then never reoccur for any larger cardinals. A similar justification is used for strongly compact cardinals. Given a \(\mathcal L_{\aleph_0,\aleph_0}\)-theory \(T\), if a \(\mathcal L_{\aleph_0,\aleph_0}\)-formula follows from \(T\), it follows from a finite subset of \\(T\\). The only infinite cardinals with this property are either \(\aleph_0\) or strongly compact cardinals, and if one wants to avoid the phenomenon stopping permanently after \(\aleph_0\), the existence of strongly compact cardinals must be assumed. Barwise recounts about the above:<ref>J. Barwise, ''Admissible Sets and Structures'', p.364. Perspectives in Mathematical Logic (1975).</ref> : The remarkable argument that strongly compact cardinals exist "by analogy with \(\omega\)" always reminds me of the goofang, described in ''The Book of Imaginary Beings'', by Jorge Luis Borges: :: The yarns and tall tales of the lumber camps of Wisconsin and Minnesota include some singular creatures, in which, surely, no one ever believed... :: There's another fish, the Goofang, that swims backward to keep the water out of its eyes. It's described as "about the size of a sunfish, only much bigger". ==References== 5ca6ff8ec83d5aa3c45901e40c642085704ac1f2 0 172 580 435 2023-09-21T20:34:51Z C7X 9 wikitext text/x-wiki Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''' (Main System with Passthrough), is known to be ill-founded.<ref>Discord message in #taranovsky-notations</ref> == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> 0ed893960cae572101bd65c4b5cd43c8cce675d1 600 580 2023-10-22T11:35:58Z C7X 9 wikitext text/x-wiki Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''' (Main System with Passthrough), is known to be ill-founded.<ref>Discord message in #taranovsky-notations</ref> == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> == Sources == 5df3fdf76996125f2a48a5e16e4d98cf48f3078c 0 219 581 2023-09-24T14:35:59Z RhubarbJayde 25 Redirected page to [[Hilbert's Grand Hotel]] wikitext text/x-wiki #REDIRECT [[Hilbert's Grand Hotel]] fdbce90effa8840ad250f4e443cd03a905d8ea0a 0 220 582 2023-09-24T14:36:26Z RhubarbJayde 25 Redirected page to [[Hilbert's Grand Hotel]] wikitext text/x-wiki #REDIRECT [[Hilbert's Grand Hotel]] fdbce90effa8840ad250f4e443cd03a905d8ea0a 0 221 583 2023-09-24T14:36:56Z RhubarbJayde 25 Redirected page to [[Hilbert's Grand Hotel]] wikitext text/x-wiki #REDIRECT [[Hilbert's Grand Hotel]] fdbce90effa8840ad250f4e443cd03a905d8ea0a 0 198 584 503 2023-09-24T19:03:23Z RhubarbJayde 25 wikitext text/x-wiki Hilbert's Grand Hotel is a famous analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the way infinite [[Bijection|bijections]] work and the fact that they go against common sense, it is possible to still fit many more people. Firstly, if there is a single new guest who wants a room, it is possible to accommodate them like so - the hotel night manager can tell everybody to move up one room, so the person checked into room zero moves to room one, the person checked into room one moves to room two, and so on. Because the set of rooms is never-ending, we don't run out of rooms and everybody who was checked in still has a room. Yet room number zero is now empty - the new guest can check in there. This is analogous to the proof that [[Omega|\(\omega\)]] and \(\omega+1\) are equinumerous. Similarly, if someone in room \(n\) checks out of the hotel, then everybody in room \(m\) for \(m > n\) can move one room to the left, and all the rooms will be filled again. One can also accommodate countably infinitely many new guests, by requiring that every current guest in room \(n\) goes to room \(2n\) and the \(n\)th of the new guests goes to room \(2n+1\). This frees up all the odd-numbered rooms, which the new guests can fill up. Therefore, \(\omega 2\) is equinumerous with \(\omega\). In fact, it's even possible to accomodate a countably infinite collection of countably infinitely many sets of new guests! One can assign the current guest in room \(n\) to room \(2^n\), the \(n\)th guest in the first collection of new guests to room \(3^n\), the \(n\)th guest in the next collection of new guests to room \(5^n\), then \(7^n\), \(11^n\), and so on. Because there are infinitely many prime numbers, and powers of primes never overlap, everybody can be accomodated - even with many rooms now empty, such as room 6, which isn't a power of any prime number! 6065e913247f6408e2592b82c35c1f4d6a50de04 0 156 585 373 2023-09-25T16:38:20Z RhubarbJayde 25 wikitext text/x-wiki The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the [[axiom of choice]], since it implies that there is no [[Well-ordered set|well-ordering]] of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is very high. Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). 328790c03c5e7197f58c36f2815643e390ec86f6 0 112 586 461 2023-09-27T06:06:28Z C7X 9 /* History */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. Bachmann's method was extended to use higher cardinals, e.g. to use \(\Omega_n\) for all finite \(n\) by Pfeiffer in 1964 and to use \(\Omega_\alpha\) for \(\alpha<I\) by Isles in 1970,<ref>Buchholz, Feferman, Pohlers, Sieg, ''Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies''. Lecture Notes in Mathematics (1981). Springer Berlin Heidelberg, ISBN 9783540386490.</ref> but with similarly cumbersome definitions. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref>^{Maybe not? Look at things around p.11 more}<!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 84d51c3510d250a45b8a77e4bd6440a87b8c35cf 587 586 2023-09-27T06:07:27Z C7X 9 /* History */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. Bachmann's method was extended to use higher cardinals, e.g. to use \(\Omega_n\) for all finite \(n\) by Pfeiffer in 1964 and to use \(\Omega_\alpha\) for \(\alpha<I\) by Isles in 1970,<ref>Buchholz, Feferman, Pohlers, Sieg, ''Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies''. Lecture Notes in Mathematics (1981). Springer Berlin Heidelberg, ISBN 9783540386490.</ref> but with similarly cumbersome definitions.<ref name="RathjenArt" />^p.11 A modern "recast", proposed by Michael Rathjen<ref name="RathjenArt">Rathjen, Michael. "[https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]".</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref>^{Maybe not? Look at things around p.11 more}<!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal da289fc2ffea64057eed195011044b211fa9caa6 592 587 2023-10-01T18:39:31Z IDoNotExist 16 /* History */ wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions. Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \varphi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. Bachmann's method was extended to use higher cardinals, e.g. to use \(\Omega_n\) for all finite \(n\) by Pfeiffer in 1964 and to use \(\Omega_\alpha\) for \(\alpha<I\) by Isles in 1970,<ref>Buchholz, Feferman, Pohlers, Sieg, ''Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies''. Lecture Notes in Mathematics (1981). Springer Berlin Heidelberg, ISBN 9783540386490.</ref> but with similarly cumbersome definitions.<ref name="RathjenArt" />^p.11 A modern "recast", proposed by Michael Rathjen<ref name="RathjenArt">Rathjen, Michael. "[https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]".</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref>^{Maybe not? Look at things around p.11 more}<!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 71b715129394184c3666e709347beed94a620634 0 103 588 295 2023-09-30T13:09:36Z RhubarbJayde 25 wikitext text/x-wiki <nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "</nowiki>[[Recursive ordinal|recursive]]<nowiki> ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning there must be some ordinals which are still countable but they aren't recursive - i.e: they're so large that all well-orders they code are so complex that they are uncomputable. The least such is the Church-Kleene ordinal. Note that there is still a well-order on the natural numbers with order type \( \omega_1^{\mathrm{CK}} \) that is computable with an </nowiki>[[Infinite time Turing machine|''infinite time'' Turing machine]], since they are able to solve the halting problem for ordinary Turing machines and thus diagonalize over the recursive ordinals. Also, note that given computable well-orders on the natural numbers with order types \( \alpha \) and \( \beta \), it is possible to construct computable well-orders with order-types \( \alpha + \beta \), \( \alpha \cdot \beta \) and \( \alpha^{\beta} \) and much more, meaning that the Church-Kleene ordinal is not pathological and in fact a limit ordinal, [[Epsilon numbers|epsilon number]], [[Strongly critical ordinal|strongly critical]], and more. <nowiki>It has a variety of other convenient definitions. One of them has to do with the constructible hierarchy - \( \omega_1^{\mathrm{CK}} \) is the least </nowiki>[[admissible]] ordinal. In other words, it is the least limit ordinal \( \alpha > \omega \) so that, for any \( \Delta_0(L_\alpha) \)-definable function \( f: L_\alpha \to L_\alpha \), then, for all \( x \in L_\alpha \), \( f<nowiki>''</nowiki>x \in L_\alpha \). That is, the set of constructible sets with rank at most \( \omega_1^{\mathrm{CK}} \) is closed under taking preimages of an infinitary analogue of the primitive recursive functions. Note that this property still holds for \( \Sigma_1(L_\alpha) \)-functions, an infinitary analogue of Turing-computable functions, which makes sense, since the ordinals below \( \omega_1^{\mathrm{CK}} \) are a very robust class and closed under computable-esque functions. It is, in particular, equivalent to the statement: for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point (equivalent to a fixed point) of \( f \) below \( \alpha \). This also is a good explanation, since it shows it's a limit of epsilon numbers (and thus itself an epsilon number), of strongly critical numbers, etc. and why it's greater than recursive ordinals like the [[Bird ordinal]]. However, the case with \( \Sigma_2(L_\alpha) \)-functions produces a much stronger notion known as \( \Sigma_2 \)-admissibility. <nowiki>Note that, like how there is a fine hierarchy of recursive ordinals and functions on them, there is a fine hierarchy of nonrecursive ordinals, above \( \omega_1^{\mathrm{CK}} \), arguably richer.</nowiki> <nowiki>One thing to note is that many known ordinal collapsing functions are, or should be, \( \Sigma_1(L_{\omega_1^{\mathrm{CK}}}) \)-definable. Thus, the countable collapse is actually a recursive collapse, and replacing \( \Omega \) with \( \omega_1^{\mathrm{CK}} \) in an ordinal collapsing function is a possibility. While many authors do this, since it allows them to use structure more efficiently and not assume </nowiki>[[Large cardinal|large cardinal axioms]], more cumbersome proofs would be necessary, and this has led many authors such as Rathjen to instead opt for the traditional options, or use uncountable intermediates between countable nonrecursive fine structure and large cardinals, such as the reducibility hierarchy. f51eada67d883d3e415dcb071c44fe1df15688fd 0 17 589 306 2023-09-30T13:11:17Z RhubarbJayde 25 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recurisve ordinals to theories according to the lengths of the recursive wellorders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ aa64117d657af6d0ccd3ac5776c5505c9f2bc7d2 610 589 2023-12-26T16:14:48Z RhubarbJayde 25 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 13f648cfb5fb77e9739d906bc16573ccff3524f3 0 160 590 382 2023-09-30T13:12:53Z RhubarbJayde 25 wikitext text/x-wiki A set is one of the basic objects in the mathematical discipline of set theory, upon which most of apeirology is built. Although there's no technical way to define a set, a set is usually considered a collection of objects, and visualized as a bag. For example, the bag can be empty, yielding the [[empty set]]. In some systems of set theory, one has urelements, objects which aren't sets - however, a vast majority of set theory is pure set theory where everything eventually reduces to a set. For example, the [[Von Neumann ordinal]] assignment defines the [[natural numbers]] as nested bags, with zero being empty, and taking successor being adding the number to its own bag - that is, \(0 = \emptyset\) and \(a+1 = a \cup \{a\}\). The discipline of [[set theory]] has developed various operations for constructing and comparing sets as well as rules for how sets are behaved, and what properties abstract, infinite sets such as [[Ordinal|ordinals]] have. b51dce754321dc39e138a3135b07944c614bc00f 0 222 591 2023-09-30T13:49:11Z RhubarbJayde 25 Created page with "Set theory is a branch of mathematics involving the study of [[Set|sets]]. These are collections of objects. Set theory is often used as a foundation for mathematics, as many mathematical objects (such as [[natural numbers]], groups, topological spaces, ...) can all be encoded as sets. In pure set theory, the primary objects of study are [[infinite]] sets, which include infinite [[Ordinal|ordinals]] and [[Cardinal|cardinals]]. Set theory is the basis for a lot of apeirol..." wikitext text/x-wiki Set theory is a branch of mathematics involving the study of [[Set|sets]]. These are collections of objects. Set theory is often used as a foundation for mathematics, as many mathematical objects (such as [[natural numbers]], groups, topological spaces, ...) can all be encoded as sets. In pure set theory, the primary objects of study are [[infinite]] sets, which include infinite [[Ordinal|ordinals]] and [[Cardinal|cardinals]]. Set theory is the basis for a lot of apeirology, as it provides a basis for formalising or defining apeirological concepts, and has introduced useful techniques for proof such as Mostowski collapse, or Skolem hull. f3246858b83694bafaacb8cd1f224448909ecec5 611 591 2023-12-26T16:16:22Z RhubarbJayde 25 wikitext text/x-wiki Set theory is a branch of mathematics involving the study of [[Set|sets]]. These are collections of objects. Set theory is often used as a foundation for mathematics, as many mathematical objects (such as [[natural numbers]], groups, topological spaces, ...) can all be encoded as sets. In pure set theory, the primary objects of study are [[infinite]] sets, which include infinite [[Ordinal|ordinals]] and [[Cardinal|cardinals]]. Set theory is the basis for a lot of apeirology, as it provides a basis for formalising or defining apeirological concepts, and has introduced useful techniques for proof such as Mostowski collapse, or Skolem hull. Set theory is very broad, and is comprised of many other subjects such as [[proof theory]] and [[model theory]] fc45f2fcd5fe910c6d612368c39ebd2b70f8f77f 612 611 2023-12-26T16:17:51Z RhubarbJayde 25 wikitext text/x-wiki Set theory is a branch of mathematics involving the study of [[Set|sets]]. These are collections of objects. Set theory is often used as a foundation for mathematics, as many mathematical objects (such as [[natural numbers]], groups, topological spaces, ...) can all be encoded as sets. In pure set theory, the primary objects of study are [[infinite]] sets, which include infinite [[Ordinal|ordinals]] and [[Cardinal|cardinals]]. Set theory is the basis for a lot of apeirology, as it provides a basis for formalising or defining apeirological concepts, and has introduced useful techniques for proof such as Mostowski collapse, or Skolem hull. Set theory is very broad, and is comprised of many other subjects such as [[proof theory]] and [[model theory]]. 90ebe0b68b6e2820a9c6468c68ee4a01d859d7c5 0 121 593 510 2023-10-02T05:54:42Z C7X 9 /* Levy-Montague reflection */ wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>https://arxiv.org/abs/1708.06669</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-[[Correct cardinal|correct]]. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give sine informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. ==References== ee22beedab04b502e8aae7848f6b736ca374a4a5 0 223 594 2023-10-02T06:36:09Z C7X 9 Created page with "A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x)\)<ref>, then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). [https://logicdavid.github.io/files/mthes..." wikitext text/x-wiki A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x)\)<ref>, then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). [https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> eb43fd5d30329ca3d305edac531ad8dbd59854ec 595 594 2023-10-02T06:38:48Z C7X 9 wikitext text/x-wiki A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>[https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> e038092f15a68c63b1306a24e9c31b632f26290f 596 595 2023-10-02T06:39:51Z C7X 9 wikitext text/x-wiki A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>[https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> bf93f321fc5b993c8f9d969d1a26ca761b4d7919 2 224 597 2023-10-06T17:21:36Z Alemagno12 27 Created page with "Here are some (conjectured) translation maps between SSS and BM2.3. Term(PrSS), Term(PSS), Term(TSS), Term(SSS) are the sets of all PrSS, PSS, TSS and SSS expressions respectively. ES is the empty string/matrix, + is string/matrix concatenation, and > is lexicographic comaparison. == Up to \(\varepsilon_0\) == Define a function Prune : Term(SSS) -> Term(SSS) inductively as follows: - Prune(ES) = ES - Prune((A,0)) = (Prune(A),0) - Prune((A,0,B,1)) = (Prune(A),0,B), whe..." wikitext text/x-wiki Here are some (conjectured) translation maps between SSS and BM2.3. Term(PrSS), Term(PSS), Term(TSS), Term(SSS) are the sets of all PrSS, PSS, TSS and SSS expressions respectively. ES is the empty string/matrix, + is string/matrix concatenation, and > is lexicographic comaparison. == Up to \(\varepsilon_0\) == Define a function Prune : Term(SSS) -> Term(SSS) inductively as follows: - Prune(ES) = ES - Prune((A,0)) = (Prune(A),0) - Prune((A,0,B,1)) = (Prune(A),0,B), where B >= 1 - Otherwise, Prune(A) doesn't exist. Define a function Shift : Term(PrSS) -> Term(PrSS) inductively as follows: - Shift(ES) = ES - Shift((A,k)) = (Shift(A),k+1) i.e. Shift(A) is the result of adding 1 to all the elements in A. Define a function Map : Term(SSS) -> Term(PrSS) inductively as follows: - Map(ES) = ES - Map((A,0)) = (Map(A),0) - Map((0,A)) = I'll finish this later, also maybe a more algorithmic approach would be better? E.g first turn all (0,1^k)s into (k)s, then add inbetween elements afb927e7828017da10d5239e6aeab0ee5e610b82 598 597 2023-10-06T17:22:07Z Alemagno12 27 Formatting wikitext text/x-wiki Here are some (conjectured) translation maps between SSS and BM2.3. Term(PrSS), Term(PSS), Term(TSS), Term(SSS) are the sets of all PrSS, PSS, TSS and SSS expressions respectively. ES is the empty string/matrix, + is string/matrix concatenation, and > is lexicographic comaparison. == Up to \(\varepsilon_0\) == Define a function Prune : Term(SSS) -> Term(SSS) inductively as follows: * Prune(ES) = ES * Prune((A,0)) = (Prune(A),0) * Prune((A,0,B,1)) = (Prune(A),0,B), where B >= 1 * Otherwise, Prune(A) doesn't exist. Define a function Shift : Term(PrSS) -> Term(PrSS) inductively as follows: * Shift(ES) = ES * Shift((A,k)) = (Shift(A),k+1) i.e. Shift(A) is the result of adding 1 to all the elements in A. Define a function Map : Term(SSS) -> Term(PrSS) inductively as follows: * Map(ES) = ES * Map((A,0)) = (Map(A),0) * Map((0,A)) = I'll finish this later, also maybe a more algorithmic approach would be better? E.g first turn all (0,1^k)s into (k)s, then add inbetween elements 862006ab87c53cb2626202d6ecabef0454fd26d9 608 598 2023-11-13T05:41:02Z Alemagno12 27 wikitext text/x-wiki Here are some translation maps between SSS and BM2.3, alongside their proofs. For a BM2.3 or SSS expression X, o_BM2.3(X) or o_SSS(X) is the order type of all possible standard expressions below it under expansion; for BM2.3, this is guaranteed to be an ordinal by [https://arxiv.org/abs/2307.04606 <nowiki>[Yto2023]</nowiki>]. ES is the empty string/matrix, + is string/matrix concatenation, X*k = X+X+X+...+X k times for some natural number k, and > is ordering by expansion. TODO: Prove that lexicographic ordering is equivalent to expansion for SSS. This is already proven for BM2.3 in <nowiki>[Yto2023]</nowiki> == Up to \(\varepsilon_0\) == <b>Lemma 1.</b> Assume that o_SSS(X) = o_BM2.3(X') and o_SSS(Y) = o_BM2.3(Y') for some standard X,X',Y,Y'. Then, o_SSS(X+(0)+Y) = o_BM2.3(X'+Y'). <i>Proof.</i> By induction on Y. Zero and successor cases are trivial. For the limit case, let Z < Y and o_SSS(Z) = o_BM2.3(Z'). * BM2.3 does not have any "stop at the leftmost element if you can't find the bad root" condition, so the bad root of X'+Z' is the bad root of Z', and o_BM2.3(X'+Z') = sup_n<ω(o_BM2.3(X'+exp(Z',n))). * For SSS, if the second bad root check finds a bad root in Z, the bad root of X+(0)+Z is the bad root of Z. If it does not find a bad root, then since the first bad root must be at least (0,1) and every standard expression starts with 0, (0)+Z = (0,0)+W for some W < (0,1) <= bad root; WIP 7b1ee4eeb8eccd0b8caf16a99112bc1bf893513f 0 85 599 420 2023-10-21T03:38:19Z C7X 9 wikitext text/x-wiki The '''large Veblen ordinal''', also called the '''great Veblen number''',<ref>Rathjen, https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, p.10</ref> is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,\ldots,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi\begin{pmatrix} 1 \\ \alpha \end{pmatrix} \), where \( \begin{pmatrix} 1 \\ \alpha \end{pmatrix} \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi\begin{pmatrix} 1 \\ (1,0) \end{pmatrix} \) (should there be parentheses in the second row?), which represents the large Veblen ordinal.<ref>https://arxiv.org/abs/2310.12832v1</ref> This system's limit is the [[Bachmann-Howard ordinal]]. 686d36af54883930b4af05035da8d3a6e36f470a 0 177 601 444 2023-10-22T11:37:33Z C7X 9 {0,1} used later wikitext text/x-wiki The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(0,1),(0,2),(1,2),(1,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. In particular, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, the cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. c8f8894298d43ca1f3ee10ec892d94ab6e760abb 602 601 2023-10-22T11:37:54Z C7X 9 Undo revision 601 by [[Special:Contributions/C7X|C7X]] ([[User talk:C7X|talk]]) wikitext text/x-wiki The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. In particular, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, the cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. 211f17d63d1182916695efb99f21371072ebb1b0 609 602 2023-12-22T22:35:20Z 80.192.118.48 0 keeping the wiki alive wikitext text/x-wiki The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. Particularly, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, the cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. d2f9119bbbc08d58971188878ff1808158a0abe0 3 225 604 2023-10-22T23:21:19Z C7X 9 /* Subdivision candidates */ new section wikitext text/x-wiki == Subdivision candidates == Some ideas for conditions for refining the hierarchy of ordinals * \(\Sigma_1\land\Pi_1\) conditions were mentioned on GS, the idea coming from determinacy. \(\Sigma_1\land\Pi_1\) formula include the \(\Sigma_1\) formulae by \(\Sigma_1\land\top\) and the \(\Pi_1\) formulae by \(\top\land\Pi_1\), and every \(\Sigma_1\land\Pi_1\) formula is itself \(\Delta_2\). * An Ershov hierarchy but for \(\Delta_2(L_\alpha)\) sets. The usual Ershov hierarchy has \(\mathcal D_n\) consist of the sets which are symmetric differences of \(n\) recursively enumerable sets, maybe this can be done for \(\Sigma_1(L_\alpha)\) sets (e.g. "\(\mathcal D_2\) but for \(L_\alpha\)" will be the sets definable by \(\Sigma_1\oplus\Sigma_1\) formulas.) The Ershov hierarchy has the benefit of [https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/on-transfinite-levels-of-the-ershov-hierarchy/96FCFB2D3989682C95CA1387364A6B58 transfinite extensions existing], for example here ("[https://homepages.ecs.vuw.ac.nz/~melnikal/sigma03beyond.pdf On a difference hierarchy for arithmetical sets]") is another extension with \(\varepsilon_0\) levels that includes all arithmetical sets. * \(\Sigma_1(St)\) formulae. * To go below admissibility, the \(\{f\mid f\textrm{ is a restriction of }(\gamma,\delta)\mapsto\gamma+\delta\}\)-cardinals are the additively indecomposable \(\alpha\). There may have to be an \(\alpha>0\) requirement throughout this post. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 23:21, 22 October 2023 (UTC) a42f22dc4fbcfd56c0fbca885b874f3bed23067f 605 604 2023-10-22T23:23:48Z C7X 9 /* Subdivision candidates */ wikitext text/x-wiki == Subdivision candidates == Some ideas for conditions for refining the hierarchy of ordinals * \(\Sigma_1\land\Pi_1\) conditions were mentioned on GS, the idea coming from determinacy. \(\Sigma_1\land\Pi_1\) formula include the \(\Sigma_1\) formulae by \(\Sigma_1\land\top\) and the \(\Pi_1\) formulae by \(\top\land\Pi_1\), and every \(\Sigma_1\land\Pi_1\) formula is itself \(\Delta_2\). * An Ershov hierarchy but for \(\Delta_2(L_\alpha)\) sets. The usual Ershov hierarchy has \(\mathcal D_n\) consist of the sets which are symmetric differences of \(n\) recursively enumerable sets, maybe this can be done for \(\Sigma_1(L_\alpha)\) sets (e.g. "\(\mathcal D_2\) but for \(L_\alpha\)" will be the sets definable by \(\Sigma_1\oplus\Sigma_1\) formulas.) The Ershov hierarchy has the benefit of [https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/on-transfinite-levels-of-the-ershov-hierarchy/96FCFB2D3989682C95CA1387364A6B58 transfinite extensions existing], for example here ("[https://homepages.ecs.vuw.ac.nz/~melnikal/sigma03beyond.pdf On a difference hierarchy for arithmetical sets]") is another extension with \(\varepsilon_0\) levels that includes all arithmetical sets. * \(\Sigma_1(St)\) formulae. * To go below admissibility, the \(\{f\mid f\textrm{ is a restriction of }(\gamma,\delta)\mapsto\gamma+\delta\}\)-cardinals are the additively indecomposable \(\alpha\), this isn't very good but maybe something can come from the [https://mathoverflow.net/q/388619 primitive recursive ordinal functions]. There may have to be an \(\alpha>0\) requirement throughout this post. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 23:21, 22 October 2023 (UTC) 2021c8c36b7287d4546f740521521b16487ae2ef 606 605 2023-10-23T01:09:34Z C7X 9 /* Subdivision candidates */ wikitext text/x-wiki == Subdivision candidates == Some ideas for conditions for refining the hierarchy of ordinals * \(\Sigma_1\land\Pi_1\) conditions were mentioned on GS, the idea coming from determinacy. \(\Sigma_1\land\Pi_1\) formula include the \(\Sigma_1\) formulas by \(\Sigma_1\land\top\) and the \(\Pi_1\) formulas by \(\top\land\Pi_1\), and every \(\Sigma_1\land\Pi_1\) formula is itself \(\Delta_2\). * An Ershov hierarchy but for \(\Delta_2(L_\alpha)\) sets. The usual Ershov hierarchy has \(\mathcal D_n\) consist of the sets which are symmetric differences of \(n\) recursively enumerable sets, maybe this can be done for \(\Sigma_1(L_\alpha)\) sets (e.g. "\(\mathcal D_2\) but for \(L_\alpha\)" will be the sets definable by \(\Sigma_1\oplus\Sigma_1\) formulas on \(L_\alpha\).) The Ershov hierarchy has the benefit of [https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/on-transfinite-levels-of-the-ershov-hierarchy/96FCFB2D3989682C95CA1387364A6B58 transfinite extensions existing], for example here ("[https://homepages.ecs.vuw.ac.nz/~melnikal/sigma03beyond.pdf On a difference hierarchy for arithmetical sets]") is another extension with \(\varepsilon_0\) levels that includes all arithmetical sets. formulas. * To go below admissibility, the \(\{f\mid f\textrm{ is a restriction of }(\gamma,\delta)\mapsto\gamma+\delta\}\)-cardinals are the additively indecomposable \(\alpha\), this isn't very good but maybe something can come from the [https://mathoverflow.net/q/388619 primitive recursive ordinal functions]. There may have to be an \(\alpha>0\) requirement throughout this post. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 23:21, 22 October 2023 (UTC) a3db3283ca2c7d460799756e41ef5e21fddbfe98 607 606 2023-10-23T20:48:17Z C7X 9 wikitext text/x-wiki == Subdivision candidates == Some ideas for conditions for refining the hierarchy of ordinals * \(\Sigma_1\land\Pi_1\) conditions were mentioned on GS, the idea coming from determinacy. \(\Sigma_1\land\Pi_1\) formula include the \(\Sigma_1\) formulas by \(\Sigma_1\land\top\) and the \(\Pi_1\) formulas by \(\top\land\Pi_1\), and every \(\Sigma_1\land\Pi_1\) formula is itself \(\Delta_2\). * An Ershov hierarchy but for \(\Delta_2(L_\alpha)\) sets. The usual Ershov hierarchy has \(\mathcal D_n\) consist of the sets which are symmetric differences of \(n\) recursively enumerable sets, maybe this can be done for \(\Sigma_1(L_\alpha)\) sets (e.g. "\(\mathcal D_2\) but for \(L_\alpha\)" will be the sets definable by \(\Sigma_1\oplus\Sigma_1\) formulas on \(L_\alpha\).) The Ershov hierarchy has the benefit of [https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/on-transfinite-levels-of-the-ershov-hierarchy/96FCFB2D3989682C95CA1387364A6B58 transfinite extensions existing], for example here ("[https://homepages.ecs.vuw.ac.nz/~melnikal/sigma03beyond.pdf On a difference hierarchy for arithmetical sets]") is another extension with \(\varepsilon_0\) levels that includes all arithmetical sets. * \(\Sigma_1(St)\) formulas. * To go below admissibility, the \(\{f\mid f\textrm{ is a restriction of }(\gamma,\delta)\mapsto\gamma+\delta\}\)-cardinals are the additively indecomposable \(\alpha\), this isn't very good but maybe something can come from the [https://mathoverflow.net/q/388619 primitive recursive ordinal functions]. There may have to be an \(\alpha>0\) requirement throughout this post. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 23:21, 22 October 2023 (UTC) af6b40d1e3bdea8f53e81b8253d6c2a1e80e2a4d 0 15 613 314 2023-12-26T16:23:15Z RhubarbJayde 25 Removed a link wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by BashicuHyudora. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs). Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> 05649e5670ad068cff1b405f5ea7fb0c8c159432 0 226 614 2023-12-26T16:25:37Z RhubarbJayde 25 Created page with "Nothing OCF is a weak [[Ordinal collapsing function|OCF]], defined by CatIsFluffy. It is similar to many other OCFs in definition, but omits addition. Therefore, the growth rate is much, much slower. It is believed to correspond to a weak version of [[Extended Buchholz's function]], also defined by omitting addition, and that it catches up to the ordinary version of EBOCF by [[Extended Buchholz ordinal|EBO]]. However, no proof of either of these claims has been given and..." wikitext text/x-wiki Nothing OCF is a weak [[Ordinal collapsing function|OCF]], defined by CatIsFluffy. It is similar to many other OCFs in definition, but omits addition. Therefore, the growth rate is much, much slower. It is believed to correspond to a weak version of [[Extended Buchholz's function]], also defined by omitting addition, and that it catches up to the ordinary version of EBOCF by [[Extended Buchholz ordinal|EBO]]. However, no proof of either of these claims has been given and it remains an open question. fcbb2cc15c3833c923153b382c72bf19e3896d90 0 226 615 614 2023-12-26T16:25:51Z RhubarbJayde 25 wikitext text/x-wiki Nothing OCF is a weak [[Ordinal collapsing function|OCF]], defined by CatIsFluffy. It is similar to many other OCFs in definition, but omits addition. Therefore, the growth rate is much, much slower. It is believed to correspond to a weak version of [[Extended Buchholz's function]], also defined by omitting addition, and that it catches up to the ordinary version of EBOCF by [[Extended Buchholz ordinal|EBO]]. However, no proof of either of these claims has been given and they remain open questions. fefcbce2a067ab3c9135b9b2763a46590cbe5c59 0 54 616 324 2023-12-26T16:27:15Z RhubarbJayde 25 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu: \textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. == Extension == This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. == References == 5b09ab06d4fc873bda68afa7722c00d4ad7199d6 617 616 2023-12-26T16:27:42Z RhubarbJayde 25 wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu: \textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_\mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. == Extension == This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_\mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. == References == 877c52fbc37086276c727c21232a7d9c784f2f99 0 193 618 487 2023-12-26T16:29:24Z RhubarbJayde 25 wikitext text/x-wiki Cantor's diagonal argument is a method for showing the [[Countability|uncountability]] of the set of real numbers. It is a proof by contradiction - one assumes that, towards contradiction, there is a [[bijection]] from the [[natural numbers]] to the real numbers, and then one constructs a real number not in the range of this function, which contradicts surjectivity. It may be rephrased as the assertion that every function from the naturals to the reals is non-surjective, and in that sense it could be considered a direct proof instead. First, for a real number \(r\), let \(r[n]\) be the \(n\)th digit after the decimal place, in the "lexicographically maximal" decimal expansion of \(r\). For example, \(\pi[0] = 1\) and \(\pi[1] = 4\). Then the proof goes like so: assume \(f: \mathbb{N} \to \mathbb{R}\) is a supposed function enumerating the entirety of the real numbers. Let \(r\) be the real number with whole part 0. Then \(r[n] = 0\) if \(f(n)[n] \neq 0\), and else \(r[n] = 1\). Now, since \(f\) is surjective, there is some \(m\) so that \(f(m) = r\). Then, if \(f(m)[m] \neq 0\), we have \(r[m] = 0\), and if \(f(m)[m] = 0\), we have \(r[m] = 1\). That is, \(r\) and \(f(m)\) disagree on the \(m\)th decimal place after the decimal point, and therefore they can not be the same real number. The proof is known as the "diagonal argument" because of a common visual representation. Namely, one writes out the decimal expansions of all the supposed real numbers on a grid, so that \(f(n)[m]\) is equal to the entry at the \(m+1\)st column and \(n+1\)st row, and then constructs a new real number by "inverting" the decimal expansion of the diagonal. The conclusion can be strengthened to show that just the real interval \([0,1]\) is uncountable. == P(N) == A similar argument can be used to show the uncountability of the [[powerset]] of the natural numbers, \(\mathcal{P}(\mathbb{N})\). Note that, by using binary expansions and characteristic functions, one can see that \([0,1]\) has the same size as \(\mathcal{P}(\mathbb{N})\), with \(0\) corresponding to [[Empty set|\(\emptyset\)]] and \(1\) corresponding to \(\mathbb{N}\). However, the application of the method of diagonalization to \(\mathcal{P}(\mathbb{N})\) has some merit in its own right. Assume \(f: \mathbb{N} \to \mathcal{P}(\mathbb{N})\) is a supposed function enumerating the entirety of the powerset of the natural numbers. We define a set \(X\) like so: \(n \in X\) iff \(n \notin f(n)\). Now, since \(f\) is surjective, there is some \(m\) so that \(f(m) = X\). Then, we have \(m \in X\) iff \(m \notin f(m)\) iff \(m \notin X\). This is impossible! And this can be generalized to show that, for every set \(X\), we have \(2^{|X|} > |X|\). The general form of diagonalization-type arguments is similar to the barber paradox, or, more precisely, Russel's paradox. e66859e3144ae519c9229db50e4d2f707792b09f 0 143 619 544 2024-01-18T04:43:21Z C7X 9 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (\(L\) does not have the [[covering property]]).<ref>Any text about Jensen's covering theorem</ref> * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref> * There is a proper class of nontrivial elementary embeddings \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\).<ref name=":0">W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref> * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\).<ref name=":0" /> While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable (stable for first-order formulae?) - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. Alternatively, \(0^\sharp\) may be defined as a sound mouse (iterable premouse), or as an Ehrenfeucht-Mostowski blueprint. dd1eecab582b061f728698cefa7827199fddbc9d 0 17 620 610 2024-01-18T04:46:08Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 9432ee8169d5202adeee217c93ae5e1cdd529829 621 620 2024-01-18T05:04:44Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 36351d7a370902f05894ad38808167abba954fc1 622 621 2024-01-18T05:10:32Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension</nowiki> * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 355d2f7f49b541bdc1892f74e5a1f27f63355ac5 623 622 2024-01-18T06:27:50Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = = \(\Pi^1_2\)-ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Pi^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ f889b76401facd45ff560af8cec4d01d4eba1032 624 623 2024-01-18T06:31:35Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 907b445e515b5a42e8049e5fd14fbe69d00ffc2d 625 624 2024-01-18T06:36:52Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 65c17686efcdc0d2f794d86e8fc1a681dccfc6ec 626 625 2024-01-18T08:05:56Z C7X 9 Possible better searchability wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its existence), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 5ee8ccb5b4b6bf52c6aacc895e6d39d243458196 627 626 2024-01-18T10:28:37Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 7be7fb6dcfb8cac2783d02a95ce8b43d4690882b 628 627 2024-01-18T10:45:57Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ d5e9354482442f8bc6e0b3b1962c2db84b2ba5a7 629 628 2024-01-18T10:50:53Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ e43b9db5a4b38b04fe53694cba4126a0bd453e36 630 629 2024-01-19T07:07:58Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ a29d8fb046f5447082bcfe2b47444c35c08882d7 631 630 2024-01-19T07:15:08Z C7X 9 Projective determinacy, Σ^1_(n+2)-rfl. and Π^1_(n+2)-rfl. wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for <math>\alpha<\omega_1^{M_n}</math>, <math>\alpha</math> is <math>M_n</math>-stable iff it is <math>\Sigma^1_{n+2}</math>-reflecting when <math>n</math> is even, and <math>\Pi^1_{n+2}</math>-reflecting when <math>n</math> is odd.<ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ a95ef6c5f3ce1e4082e0e13009b8513b37978d70 632 631 2024-01-19T07:18:23Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)}<ref name="OrderOfReflection" />^(p.20) * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 09fcf756c335b02c3f635838572308a55b8b9e5e 633 632 2024-01-24T10:08:15Z C7X 9 Justification wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \(\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ de4229538e4f6b3a5c1de405773ebb6a273b81ba 634 633 2024-01-24T10:08:37Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 64b051a2bd5997b407a1ea197de3ec7ad3a15ed1 635 634 2024-01-24T11:36:05Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ c4343ddfdf7dd26e5e03a4e89205677ff4231069 636 635 2024-01-26T05:58:12Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀ * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 1a4a63b71e42f30d8cf1fdfa2d4739700bdaf6e2 637 636 2024-01-30T23:47:21Z C7X 9 Seems unusual to see "L_(recursive ordinal) n P(omega) models (a theory of second-order arithmetic)" wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_\eta\), for \(\eta > 0\)<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 478d40f735459b39e833f015b838abade1a57cb9 638 637 2024-02-02T02:42:13Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\).<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> In unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\).<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'' (2010?, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ e9770cbe6a7d72a5b7b3e151d249eeb0fdbde1df 639 638 2024-02-02T02:43:15Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\).<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'' (2010?, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ e255b9ba91488c5f49235386954fc9e18390bced 640 639 2024-02-02T02:43:30Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'' (2010?, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 449a482d14fb2ce86229ad7c6dff8cb97c6b6236 641 640 2024-02-02T02:44:13Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'', edited by R. Schindler, Ontos Series in Mathematical Logic (2010, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" /> * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 7e0b8644ccbbced6fe3a4ddd807cde28f7dd145e 642 641 2024-02-02T04:33:58Z C7X 9 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α) * [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is Extended Buchholz unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \) *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'', edited by R. Schindler, Ontos Series in Mathematical Logic (2010, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies \(\mathsf{AQI}\), arithmetical quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" />{{verification failed}} * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 9f70cde7e39e4da300ceb599bf63fc6d2431668f 0 121 643 593 2024-02-24T19:59:15Z C7X 9 wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, J. Reitz, R. Schindler, [https://arxiv.org/abs/1708.06669 Inner-model reflection principles] (2018). Accessed 4 September 2023.</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-[[Correct cardinal|correct]]. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give sine informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. ==References== df326cdf3ca9a50fdbd0a63ad8f95224aad566a8 0 103 644 588 2024-03-16T01:12:47Z C7X 9 Correction wikitext text/x-wiki <nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "</nowiki>[[Recursive ordinal|recursive]]<nowiki> ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning there must be some ordinals which are still countable but they aren't recursive - i.e: they're so large that all well-orders they code are so complex that they are uncomputable. The least such is the Church-Kleene ordinal. Note that there is still a well-order on the natural numbers with order type \( \omega_1^{\mathrm{CK}} \) that is computable with an </nowiki>[[Infinite time Turing machine|''infinite time'' Turing machine]], since they are able to compute whether a given Turing machine computes a well-order or not,<ref>Hamkins Lewis 2000, theorems that mention "\(\mathrm{WO}\)"</ref>. Also, note that given computable well-orders on the natural numbers with order types \( \alpha \) and \( \beta \), it is possible to construct computable well-orders with order-types \( \alpha + \beta \), \( \alpha \cdot \beta \) and \( \alpha^{\beta} \) and much more, meaning that the Church-Kleene ordinal is not pathological and in fact a limit ordinal, [[Epsilon numbers|epsilon number]], [[Strongly critical ordinal|strongly critical]], and more. <nowiki>It has a variety of other convenient definitions. One of them has to do with the constructible hierarchy - \( \omega_1^{\mathrm{CK}} \) is the least </nowiki>[[admissible]] ordinal. In other words, it is the least limit ordinal \( \alpha > \omega \) so that, for any \( \Delta_0(L_\alpha) \)-definable function \( f: L_\alpha \to L_\alpha \), then, for all \( x \in L_\alpha \), \( f<nowiki>''</nowiki>x \in L_\alpha \). That is, the set of constructible sets with rank at most \( \omega_1^{\mathrm{CK}} \) is closed under taking preimages of an infinitary analogue of the primitive recursive functions. Note that this property still holds for \( \Sigma_1(L_\alpha) \)-functions, an infinitary analogue of Turing-computable functions, which makes sense, since the ordinals below \( \omega_1^{\mathrm{CK}} \) are a very robust class and closed under computable-esque functions. It is, in particular, equivalent to the statement: for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point (equivalent to a fixed point) of \( f \) below \( \alpha \). This also is a good explanation, since it shows it's a limit of epsilon numbers (and thus itself an epsilon number), of strongly critical numbers, etc. and why it's greater than recursive ordinals like the [[Bird ordinal]]. However, the case with \( \Sigma_2(L_\alpha) \)-functions produces a much stronger notion known as \( \Sigma_2 \)-admissibility. <nowiki>Note that, like how there is a fine hierarchy of recursive ordinals and functions on them, there is a fine hierarchy of nonrecursive ordinals, above \( \omega_1^{\mathrm{CK}} \), arguably richer.</nowiki> <nowiki>One thing to note is that many known ordinal collapsing functions are, or should be, \( \Sigma_1(L_{\omega_1^{\mathrm{CK}}}) \)-definable. Thus, the countable collapse is actually a recursive collapse, and replacing \( \Omega \) with \( \omega_1^{\mathrm{CK}} \) in an ordinal collapsing function is a possibility. While many authors do this, since it allows them to use structure more efficiently and not assume </nowiki>[[Large cardinal|large cardinal axioms]], more cumbersome proofs would be necessary, and this has led many authors such as Rathjen to instead opt for the traditional options, or use uncountable intermediates between countable nonrecursive fine structure and large cardinals, such as the reducibility hierarchy. a74341986b31655cc0b8d3e3e2aab1658eaf344b 2 227 645 2024-03-22T22:58:28Z CreeperBomb 30 Created page with "[https://googology.miraheze.org/wiki/User:CreeperBomb Go here for user page] [https://googology.miraheze.org/wiki/User_talk:CreeperBomb Go here for talk page]" wikitext text/x-wiki [https://googology.miraheze.org/wiki/User:CreeperBomb Go here for user page] [https://googology.miraheze.org/wiki/User_talk:CreeperBomb Go here for talk page] 5d024c26b25d8d00251ac425cdf00a222bd7cc9d 0 107 646 272 2024-03-22T23:02:04Z CreeperBomb 30 wikitext text/x-wiki {{DISPLAYTITLE:\( \omega^\omega \)}} The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordinal of [[Second-order arithmetic|\(\mathrm{RCA}_0\)]] * The proof-theoretic ordinal of [[Second-order arithmetic|\(\mathrm{WKL}_0\)]]. * The proof-theoretic ordinal of Peano arithmetic, with induction restricted to \( \Sigma^0_1 \)-formulae. * The proof-theoretic ordinal of primitive recursive arithmetic. f555db766eccab7f14ff4b348069073a8bc105f7 0 52 647 255 2024-03-22T23:03:23Z CreeperBomb 30 wikitext text/x-wiki {{DISPLAYTITLE:\( \omega^2 \)}} The ordinal \( \omega^2 \) is the least ordinal which is a limit of limit ordinals, as well as the second infinite additive principal ordinal. It is also equal to the proof-theoretic ordinal of rudimentary function arithmetic, and of Peano arithmetic with induction restricted to \( \Delta^0_0 \)-formulae. It is also the approximate growth rate of Conway's chained arrows in the fast-growing hierarchy. ca6e7f43a7a3eca8366e65aadf61b9558cf4fb24 0 102 648 243 2024-03-22T23:06:47Z CreeperBomb 30 Changed redirect target from [[Strongly critical ordinal]] to [[Veblen hierarchy]] wikitext text/x-wiki #REDIRECT [[Veblen hierarchy]] a1bf6a262c2a83d5207d304c4373ff62d9a15a96 0 55 649 297 2024-03-22T23:07:09Z CreeperBomb 30 Changed redirect target from [[Uncountable]] to [[Countability]] wikitext text/x-wiki #REDIRECT [[Countability]] 83755c693a20bad68152328fe33a6068aa6e2a3f 0 198 650 584 2024-03-22T23:23:41Z CreeperBomb 30 wikitext text/x-wiki Hilbert's Grand Hotel is a famous analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the often unintuitive ways infinite [[bijections]] work, it is actually possible to still fit many more people. Firstly, if there is a single new guest who wants a room, it is possible to accommodate by simply telling everyone to move up one room - so the person checked in Room 0 moves to Room 1, the person checked in Room 1 moves to Room 2, and so on. Because every room has a room coming after it, everybody who was checked in still has a room. Yet Room 0 is now empty - the new guest can check in there. This is analogous to the proof that [[Omega|\(\omega\)]] and \(\omega+1\) are equinumerous (that is, they have the same [[cardinality]]). Similarly, if someone in room \(n\) checks out of the hotel, then everybody in room \(m\) for \(m > n\) can move one room to the left, and all the rooms will be filled again. One can also accommodate countably infinitely many new guests, by requiring that every current guest in Room \(n\) goes to Room \(2n\) and that the \(n\)th new guest go to Room \(2n+1\). The first part frees up all the odd-numbered rooms, which the new guests can fill up. Therefore, \(\omega 2\) is equinumerous with \(\omega\). In fact, it's even possible to accommodate a countably infinite collection of countably infinitely many sets of new guests! One can assign the current guest in room \(n\) to room \(2^n\), the \(n\)th guest in the first collection of new guests to room \(3^n\), the \(n\)th guest in the next collection of new guests to room \(5^n\), then \(7^n\), \(11^n\), and so on. Because there are infinitely many prime numbers, and powers of primes never overlap, everybody can be accommodated - even with many rooms now empty, such as room 6, which isn't a power of any prime number! However, not every infinite batch of guests can fit in Hilbert's Grand Hotel. If a bus brings infinitely many guests whose names are all infinite strings made up of "a" and "b", and every string has a guest, not all of the guests can fit. In fact, it's possible to pair up each name to a real number, showing that there are more real numbers than natural numbers, even though there are infinitely many of both! c37ad2299aac8aa11e79b13cb2cd129f83ed8d8a 0 16 651 366 2024-03-23T00:38:21Z CreeperBomb 30 wikitext text/x-wiki A normal function is an [[ordinal function]] that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties: * \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\) * \(f(\alpha)=\sup \{ f(\beta) | \beta < \alpha \} \) if \(\alpha\) is a [[limit ordinal]]. Veblen's fixed point lemma, which is essential for constructing the [[Veblen hierarchy]], guarantees that, not only does every normal function have a [[fixed point]], but the class of fixed points is unbounded and their enumeration function is also normal. fe83213f3197153e10cdbb6d87269f3d582b1373 0 114 652 486 2024-03-24T03:28:44Z CreeperBomb 30 wikitext text/x-wiki An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additive principal ordinal is 1 since \(0 + 0 < 1\), and all additive principal ordinals other than 1 are limit ordinals. In particular, as can be seen from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation), additive principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some ordinal \(\gamma\). As such, the second infinite additive principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additive principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of the limits of additive principal ordinals is \(\omega^{\omega^2}\). Additive principal ordinals can be generalized to multiplicative principal ordinals and exponential principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicative principal ordinals are to additive principal ordinals as additive principal ordinals are to limit ordinals. However, exponential principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the [[epsilon numbers]]. 31ac249acf6e959f0a645a566cef14a239a07e07 0 11 653 417 2024-03-24T03:35:16Z CreeperBomb 30 wikitext text/x-wiki '''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). They can also be informally visualized as \(\omega^{\omega^{\omega^{\cdots}}}\), though this represents all the epsilon numbers identically and doesn't have a formal definition. Since the function \(\alpha\rightarrow\omega^\alpha\) is continuous in the order topology, they are the same as the closure points. Using the [[Veblen hierarchy]], they are enumerated as \(\varphi(1,\alpha)\). The least epsilon number is the limit of "predicative" [[Cantor normal form]], since, as it is a closure point of base-\(\omega\) exponentiation, it can't be reached from below via base-\(\omega\) exponentiation. Additionally, in general, \(\varphi(1,\alpha+1)\) is the least ordinal that can't be reached from \(\varphi(1,\alpha)\) base-\(\omega\) exponentiation. By Veblen's fixed point lemma, the enumerating function of the epsilon numbers is normal and thus also has fixed points - these are denoted \(\varphi(2,\alpha)\) or \(\zeta_\alpha\). (Use of the letter \(\zeta\) seems a bit difficult to find, for example sometimes it is called \(\kappa_\alpha\): https://mathoverflow.net/questions/243502) By iterating Cantor normal form and the process of taking (common) fixed points, the [[Veblen hierarchy]] is formed. This induces a natural normal form, called Veblen normal form. Its limit is not \(\zeta_0\), but a much larger ordinal, denoted \(\Gamma_0\). And in general, the ordinals that can't be obtained from below via Veblen normal form are called strongly critical. They are important in ordinal analysis. __NOEDITSECTION__ <!-- Remove the section edit links --> 310c9498e66a485b71098287051cc4b6d0418686 0 50 654 158 2024-03-24T03:36:32Z CreeperBomb 30 wikitext text/x-wiki {{stub}} '''Cantor normal form''' is a standard form of writing ordinals. Cantor's normal form theorem states that every ordinal \( \alpha \) can be written uniquely as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer. When \( \alpha \) is smaller than [[Epsilon numbers|\( \varepsilon_0 \)]], the exponents \( \beta_1 \) through \( \beta_k \) are all strictly smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form an [[ordinal notation system]] for ordinals less than \( \varepsilon_0 \). 32b2541b3328f73554f241e38cf49f46122973fb 10 228 655 2024-03-24T03:37:32Z CreeperBomb 30 Created page with ":<div class="notice metadata plainlinks" id="stub">''This article is a [[:Category:Article stubs|stub]]. You can help {{SITENAME}} by [{{fullurl:{{FULLPAGENAME}}|action=edit}} expanding it].''</div> <includeonly>[[Category:Article stubs]]</includeonly><noinclude> ---- ''This template will categorize articles that include it into [[:Category:Article stubs]].'' </noinclude>" wikitext text/x-wiki :<div class="notice metadata plainlinks" id="stub">''This article is a [[:Category:Article stubs|stub]]. You can help {{SITENAME}} by [{{fullurl:{{FULLPAGENAME}}|action=edit}} expanding it].''</div> <includeonly>[[Category:Article stubs]]</includeonly><noinclude> ---- ''This template will categorize articles that include it into [[:Category:Article stubs]].'' </noinclude> 8b29cb8adfd96ccd4ce32740cf17228860018c7d 0 15 656 613 2024-03-24T03:45:32Z CreeperBomb 30 wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by BashicuHyudora. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs). Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> d63e6bec91915b916276dcdd3eed71afd1c517cf 665 656 2024-03-25T05:51:58Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by BashicuHyudora. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs). Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> 74e93f423ab276141f66f8a1c68468388956ba9e 0 202 657 522 2024-03-24T03:55:57Z CreeperBomb 30 wikitext text/x-wiki The cofinality of an [[ordinal]] \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's [[countable]], then it's exactly \(\omega\). It is easy to see that \(\mathrm{cof}(\alpha) \leq \alpha\) for all \(\alpha\), because the identity has unbounded range. Also, \(\mathrm{cof}(\mathrm{cof}(\alpha)) = \mathrm{cof}(\alpha)\), because if there is a \(\delta < \mathrm{cof}(\alpha)\) and maps \(f: \delta \to \mathrm{cof}(\alpha)\), \(g: \mathrm{cof}(\alpha) \to \alpha\) with unbounded range, then \(g \circ f: \delta \to \alpha\) also has unbounded range, contradicting minimality of \(\mathrm{cof}(\alpha)\). An ordinal is regular if it is equal to its own cofinality, else it is singular. So: * [[0]], [[1]] and [[Omega|\(\omega\)]] are regular. * All natural numbers other than \(0\) and \(1\) are singular. * All countable infinite ordinals other than \(\omega\) are singular. * \(\mathrm{cof}(\alpha)\) is regular for any \(\alpha\). Cofinality is used in the definition of [[Inaccessible cardinal|weakly inaccessible]] cardinals. ==Without choice== Citation about every uncountable cardinal being singular being consistent with ZF 4ee7cc7116d2b4050eba0fd71d6f760d27cea9a3 0 1 659 531 2024-03-25T05:44:40Z Cobsonwabag 32 grammar wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> b6611755bc64404fa4717652238050ac2449bb2b 0 57 660 356 2024-03-25T05:48:11Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] A set \(M\) is admissible if \((M,\in)\) is a model of [[Kripke-Platek set theory]]. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\), which is itself equivalent to \(\alpha > \omega\), \(\alpha\) being a limit ordinal and \(L_\alpha\) being closed under preimages of \(\alpha\)-recursively enumerable functions.<ref>Admissible Sets and Structures, Barwise, J., ''Perspectives in Logic'', Cambridge University Press.</ref> The least admissible ordinal is [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]], although some authors omit the axiom of infinity from KP and consider [[Omega|\(\omega\)]] to be admissible. bfdc58ac6d663e49154e34721165881cafa188a1 0 209 661 555 2024-03-25T05:48:28Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Extendible cardinals are a powerful [[large cardinal]] notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \(j: V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). While the latter clearly characterization connects extendibles to supercompacts, the same can not be said for the first. However, this follows from Magidor's lemma: a cardinal \(\kappa\) is supercompact iff, for all \(\lambda > \kappa\), there exist \(\bar{\kappa} < \bar{\lambda} < \kappa\) and an elementary embedding \(j: V_{\bar{\lambda}+1} \to V_{\lambda+1}\) with critical point \(\bar{\kappa}\) so that \(j(\bar{\kappa}) = \kappa\). If \(N\) is a [[Extender model|weak extender model]]<nowiki> for \(\kappa\)'s supercompactness, then, for all \(a \in V_\lambda\), the above characterisation of supercompactness holds and, for some \(\bar{a} \in V_{\bar{\lambda}}\):</nowiki> * \(j(\bar{a}) = a\). * <nowiki>\(j(N \cap V_{\bar{\lambda}}) = N \cap V_{\bar{\lambda}}\).</nowiki> * <nowiki>\(j \upharpoonright (N \cap V_{\bar{\lambda}}) \in N\).</nowiki> 9c73284dd7e83ec7ad055c8df1ee8a00314d27c3 0 164 662 394 2024-03-25T05:50:47Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] An [[ordinal]] is called a successor if it is equal to \(\alpha + 1\) for some other \(\alpha\). An ordinal \(\alpha\) is a successor if and only if \(\max \alpha\) exists. The smallest successor ordinal is [[1]], which is also the only successor ordinal to also be [[Additive principal ordinals|additively principal]]. The least ordinal that is not a successor, other than [[0]], is [[Omega|\(\omega\)]]. If \(\beta\) is successor, then \(\alpha+\beta\) is also successor for all \(\alpha\). However, multiplication and exponentiation do not have this property. 952d48cb50b0e6e9a0b948c66312662c99c29f73 0 46 663 381 2024-03-25T05:51:30Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. In particular, it is equal to the identity in the monoid of [[natural numbers]] under addition. Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). 40240320d444202cebf71fababb900904a251d0f 0 172 664 600 2024-03-25T05:51:45Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''' (Main System with Passthrough), is known to be ill-founded.<ref>Discord message in #taranovsky-notations</ref> == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> == Sources == 6054063c3ae075372fad99152c5ccc76f442b93a 0 64 666 171 2024-03-25T05:52:38Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitrarily obscure cases, of course, the proof can be arbitrarily unusual itself. However, for common special cases, there are some shared features of the proof. <h2>[[Expansion system | Expansion systems]]</h2> In an expansion system, totality translates to "for every pair of terms \( x,y \), one is reachable from the other by expansion". The order can usually easily be proven to respect a lexicographical order or some other order known to be total (i.e. one can prove \( x\prec y \) implies \( x<y \), where \( \prec \) is the order of the expansion system and \( < \) is a total order), and then it is simply about proving that, intuitively, repeated expansion is never forced to skip any specific term. <br>Then a common property that directly leads to totality is the conjunction of \( x[n]\preceq x[n+1] \), \( x[n]\prec y\preceq x[n+1]\Rightarrow x[n]\preceq y[0] \), and the statement that iterating \( [0] \) always reaches the minimum eventually, no matter what you start with. The intuitive reason why this implies totality is that if we have \( x<y\prec z \) and \( x=z[n_0][n_1]...[n_m] \), then \( x \) can be reached from \( y \) by repeating \( [0] \) until it reaches something of the form \( z[n_0][n_1]...[n_k][a] \) with \( a>n_{k+1} \), at which point \( [b] \) is used with \( b \) minimal so that this doesn't go below \( x \), and the whole process repeats. The fact that this eventually terminates follows from looking at the \( (n_0,n_1,...,n_k,a) \) that appear that way, and noticing that this decreases lexicographically as the process moves on the sequence always decreases lexicographically and its length is bounded by \( m+1 \), so this form can only be reached finitely many times, and between all that, we're only iterating \( [0] \), which is guaranteed to decrease the term as much as we need, eventually getting us to \( x \) only by expanding \( y \). Keep in mind that this is not a formal proof. <br>It is not always true that this property holds. This page is currently unfinished. 3e68fece19328f424c2cec80eb5be652cf15cc98 706 666 2024-03-25T16:45:03Z CreeperBomb 30 wikitext text/x-wiki Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitrarily obscure cases, of course, the proof can be arbitrarily unusual itself. However, for common special cases, there are some shared features of the proof. <h2>[[Expansion system | Expansion systems]]</h2> In an expansion system, totality translates to "for every pair of terms \( x,y \), one is reachable from the other by expansion". The order can usually easily be proven to respect a lexicographical order or some other order known to be total (i.e. one can prove \( x\prec y \) implies \( x<y \), where \( \prec \) is the order of the expansion system and \( < \) is a total order), and then it is simply about proving that, intuitively, repeated expansion is never forced to skip any specific term. <br>Then a common property that directly leads to totality is the conjunction of \( x[n]\preceq x[n+1] \), \( x[n]\prec y\preceq x[n+1]\Rightarrow x[n]\preceq y[0] \), and the statement that iterating \( [0] \) always reaches the minimum eventually, no matter what you start with. The intuitive reason why this implies totality is that if we have \( x<y\prec z \) and \( x=z[n_0][n_1]...[n_m] \), then \( x \) can be reached from \( y \) by repeating \( [0] \) until it reaches something of the form \( z[n_0][n_1]...[n_k][a] \) with \( a>n_{k+1} \), at which point \( [b] \) is used with \( b \) minimal so that this doesn't go below \( x \), and the whole process repeats. The fact that this eventually terminates follows from looking at the \( (n_0,n_1,...,n_k,a) \) that appear that way, and noticing that this decreases lexicographically as the process moves on the sequence always decreases lexicographically and its length is bounded by \( m+1 \), so this form can only be reached finitely many times, and between all that, we're only iterating \( [0] \), which is guaranteed to decrease the term as much as we need, eventually getting us to \( x \) only by expanding \( y \). Keep in mind that this is not a formal proof. <br>It is not always true that this property holds. This page is currently unfinished. 55136a51229ca84711bb531fead496230b69f5b7 0 196 667 495 2024-03-25T05:53:16Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in [[ZFC]]. Aleph fixed points are large in that they are unreachable from below via the aleph operator. However, it is possible that the number of real numbers is an aleph fixed point, or more. Furthermore, the least aleph fixed point has cofinality \(\omega\), which follows from the [[Normal function|normality]] of \(f(\alpha) = \aleph_\alpha\). A regular aleph fixed point is precisely a [[Inaccessible cardinal|weakly inaccessible cardinal]], and, therefore, the [[large cardinal]] hierarchy is beyond the notion of aleph fixed points, the fixed points of their enumeration, and so on, since those can all be proven to exist and are less than the least weakly inaccessible cardinal, if it exists. In most if not all [[Ordinal collapsing function|OCFs]], the collapse of the least aleph fixed point is the [[Extended Buchholz ordinal]], which is why it is sometimes alternately referred to as the OFP, although this is technically a misnomer. faf7e218bc2d6b293a265901351a01061885f323 707 667 2024-03-25T16:45:17Z CreeperBomb 30 wikitext text/x-wiki An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in [[ZFC]]. Aleph fixed points are large in that they are unreachable from below via the aleph operator. However, it is possible that the number of real numbers is an aleph fixed point, or more. Furthermore, the least aleph fixed point has cofinality \(\omega\), which follows from the [[Normal function|normality]] of \(f(\alpha) = \aleph_\alpha\). A regular aleph fixed point is precisely a [[Inaccessible cardinal|weakly inaccessible cardinal]], and, therefore, the [[large cardinal]] hierarchy is beyond the notion of aleph fixed points, the fixed points of their enumeration, and so on, since those can all be proven to exist and are less than the least weakly inaccessible cardinal, if it exists. In most if not all [[Ordinal collapsing function|OCFs]], the collapse of the least aleph fixed point is the [[Extended Buchholz ordinal]], which is why it is sometimes alternately referred to as the OFP, although this is technically a misnomer. fc7258496dc43d424971724e5fe17fdb1257bcff 0 60 668 196 2024-03-25T05:53:24Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] A '''sequence system''' is an [[ordinal notation system]] in which the terms of the notation are sequences. Typically, it is an [[expansion system]], with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> Notable sequence systems include [[Primitive sequence system]], [[Pair sequence system]], [[Sudden sequence system]], [[Bashicu matrix system]] and [[Y sequence]]. 411d9a254172835f7cdf73c585f112d0b2b04f21 708 668 2024-03-25T16:45:38Z CreeperBomb 30 wikitext text/x-wiki A '''sequence system''' is an [[ordinal notation system]] in which the terms of the notation are sequences. Typically, it is an [[expansion system]], with the expansion chosen so that \( x[n] \) is always lexicographically smaller than \( x \), and additionally, so that \( x[0] \) is \( x \) without its last element and \( x[n] \) is always a subsequence of \( x[n+1] \). If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicographical order.<ref>Generalization of the proof of lemma 2.3 in the [https://arxiv.org/abs/2307.04606 proof of well-foundedness] of [[Bashicu matrix system | BMS]]</ref> Notable sequence systems include [[Primitive sequence system]], [[Pair sequence system]], [[Sudden sequence system]], [[Bashicu matrix system]] and [[Y sequence]]. 3c9a66819a585fb4b24ad9a8ea6522b99f7edd1c 0 73 669 571 2024-03-25T05:53:33Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).<ref name="MarekSrebrny73" />^p.368 Gap ordinals are very large. This is because, if \( \alpha \) is a gap ordinal, then \( L_\alpha \cap \mathcal{P}(\omega) \) satisfies second-order arithmetic, despite not containing ''all'' subsets of \( \omega \). Therefore, if \( \alpha \) is a gap ordinal, it is admissible, recursively inaccessible, recursively Mahlo, nonprojectible, and more. However, there can still be countable gap ordinals. There is a nice analogy between gap ordinals and cardinals. Note that \( \alpha \) is a cardinal if, for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \). If \( \alpha \) is infinite, we have \( \pi \subseteq \gamma \times \alpha \subseteq \alpha \times \alpha \subseteq V_\alpha^2 \subseteq V_\alpha \) and thus \( \pi \in V_{\alpha + 1} \). Thus, "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in V_{\alpha + 1} \)" is equivalent to being a cardinal. Meanwhile, the least ordinal satisfying the similar but weaker condition "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in L_{\alpha + 1} \)" is equal to the least gap ordinal, since it's equivalent to \( L_\alpha \) satisfying separation.<ref>R.Björn Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, Volume 4, Issue 3, 1972, Pages 229-308</ref> Harvey Friedman proved that \(L_{\beta_0}\cap\mathcal P(\omega)\) does not satisfy \(\Sigma^0_5\) determinacy.<ref>A. Montalbán, R. Shore, "[https://math.berkeley.edu/~antonio/papers/Delta4Det.pdf The Limits of Determinacy in Second Order Arithmetic]", p.22 (2011). Accessed 13 September 2023.</ref> ==Longer gaps== Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha) \cap \mathcal{P}(\omega) = \emptyset\).<ref name="MarekSrebrny73" />^p.365 If such an \( \alpha \) starts a gap, then it is said to start a gap of length \( \gamma \). It is possible for \( \alpha \) to start a gap of length \( > \alpha \): for example, the least \( \alpha \) so that \( \alpha \) starts a gap of length \( \alpha^+ \) is equal to the least admissible which is not locally countable. There can also be second-order gap, and more. An ordinal \( \alpha \) is said to start an \( \eta \)th-order gap of length \( \gamma \) if \( (L_{\alpha+\gamma} \setminus L_\alpha) \cap \mathcal{P}^\eta(\omega) = \emptyset \) and, for all \( \beta < \alpha \), \((L_\alpha \setminus L_\beta) \cap \mathcal {P}^\eta(\omega) \neq \emptyset\). The least ordinal which starts a second-order gap is greater than the least \( \alpha \) which starts a first-order gap of length \( \alpha \), and more. If \( 0^\sharp \) exists, then, for any countable \( \eta\) and any \( \gamma \) at all, there is a countable \( \delta \) which starts an \( \eta \)th-order gap of length \( \gamma \). Meanwhile, if \( V = L \), then there is no countable ordinal starting a first-order gap of length \( \omega_1 \). ==Citations== d0f23fb499ece846d22960c44e06b12ea6cb970c 709 669 2024-03-25T16:45:54Z CreeperBomb 30 wikitext text/x-wiki A gap ordinal is an ordinal \(\alpha\) such that \((L_{\alpha+1}\setminus L_\alpha)\cap\mathcal P(\omega)=\varnothing\).<ref name="MarekSrebrny73">W. Marek, M. Srebrny, "[https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe]" (1973). Annals of Mathematical Logic vol. 6, pp.359--394.</ref>^p.364 An ordinal \(\alpha\) is said to start a gap if \(\alpha\) is a gap ordinal but for all \(\beta<\alpha\), \((L_\alpha\setminus L_\beta)\cap\mathcal P(\omega)\neq\varnothing\).<ref name="MarekSrebrny73" />^p.368 Gap ordinals are very large. This is because, if \( \alpha \) is a gap ordinal, then \( L_\alpha \cap \mathcal{P}(\omega) \) satisfies second-order arithmetic, despite not containing ''all'' subsets of \( \omega \). Therefore, if \( \alpha \) is a gap ordinal, it is admissible, recursively inaccessible, recursively Mahlo, nonprojectible, and more. However, there can still be countable gap ordinals. There is a nice analogy between gap ordinals and cardinals. Note that \( \alpha \) is a cardinal if, for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \). If \( \alpha \) is infinite, we have \( \pi \subseteq \gamma \times \alpha \subseteq \alpha \times \alpha \subseteq V_\alpha^2 \subseteq V_\alpha \) and thus \( \pi \in V_{\alpha + 1} \). Thus, "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in V_{\alpha + 1} \)" is equivalent to being a cardinal. Meanwhile, the least ordinal satisfying the similar but weaker condition "for all \( \gamma < \alpha \), there is no surjection \( \pi: \gamma \longrightarrow \alpha \) with \( \pi \in L_{\alpha + 1} \)" is equal to the least gap ordinal, since it's equivalent to \( L_\alpha \) satisfying separation.<ref>R.Björn Jensen, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, Volume 4, Issue 3, 1972, Pages 229-308</ref> Harvey Friedman proved that \(L_{\beta_0}\cap\mathcal P(\omega)\) does not satisfy \(\Sigma^0_5\) determinacy.<ref>A. Montalbán, R. Shore, "[https://math.berkeley.edu/~antonio/papers/Delta4Det.pdf The Limits of Determinacy in Second Order Arithmetic]", p.22 (2011). Accessed 13 September 2023.</ref> ==Longer gaps== Given any ordinal \(\gamma<\omega_1^L\), it is possible to find an \(\alpha<\omega_1^L\) such that \((L_{\alpha+\gamma}\setminus L_\alpha) \cap \mathcal{P}(\omega) = \emptyset\).<ref name="MarekSrebrny73" />^p.365 If such an \( \alpha \) starts a gap, then it is said to start a gap of length \( \gamma \). It is possible for \( \alpha \) to start a gap of length \( > \alpha \): for example, the least \( \alpha \) so that \( \alpha \) starts a gap of length \( \alpha^+ \) is equal to the least admissible which is not locally countable. There can also be second-order gap, and more. An ordinal \( \alpha \) is said to start an \( \eta \)th-order gap of length \( \gamma \) if \( (L_{\alpha+\gamma} \setminus L_\alpha) \cap \mathcal{P}^\eta(\omega) = \emptyset \) and, for all \( \beta < \alpha \), \((L_\alpha \setminus L_\beta) \cap \mathcal {P}^\eta(\omega) \neq \emptyset\). The least ordinal which starts a second-order gap is greater than the least \( \alpha \) which starts a first-order gap of length \( \alpha \), and more. If \( 0^\sharp \) exists, then, for any countable \( \eta\) and any \( \gamma \) at all, there is a countable \( \delta \) which starts an \( \eta \)th-order gap of length \( \gamma \). Meanwhile, if \( V = L \), then there is no countable ordinal starting a first-order gap of length \( \omega_1 \). ==Citations== 795e033d03fe905cf2ed6e61dd16b875c245a06a 0 83 670 218 2024-03-25T05:53:41Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The Small Veblen ordinal is the limit of a finitary, variadic extension of the [[Veblen hierarchy]]. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. [[Veblen hierarchy|strongly critical ordinals]] - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least ordinal not reachable from below via this function, namely the limit of \( \omega \), \( \varepsilon_0 \), \( \Gamma_0 \), \( \varphi(1,0,0,0) \) (the Ackermann ordinal), ... In ordinal collapsing functions, in particular Buchholz's psi function, it is considered the countable collapse of \( \Omega^{\Omega^\omega} \), and may be denoted by \( \psi_0(\Omega^{\Omega^\omega}) \). 0e786c38b066efde203bdbc3d86c43dd8d56b28a 710 670 2024-03-25T16:46:33Z CreeperBomb 30 wikitext text/x-wiki The Small Veblen ordinal is the limit of a finitary, variadic extension of the [[Veblen hierarchy]]. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. [[Veblen hierarchy|strongly critical ordinals]] - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least ordinal not reachable from below via this function, namely the limit of \( \omega \), \( \varepsilon_0 \), \( \Gamma_0 \), \( \varphi(1,0,0,0) \) (the Ackermann ordinal), ... In ordinal collapsing functions, in particular Buchholz's psi function, it is considered the countable collapse of \( \Omega^{\Omega^\omega} \), and may be denoted by \( \psi_0(\Omega^{\Omega^\omega}) \). dda9cccfb2ba37bc3034ca406a9644badda200b3 0 103 671 644 2024-03-25T05:53:52Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] <nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "</nowiki>[[Recursive ordinal|recursive]]<nowiki> ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning there must be some ordinals which are still countable but they aren't recursive - i.e: they're so large that all well-orders they code are so complex that they are uncomputable. The least such is the Church-Kleene ordinal. Note that there is still a well-order on the natural numbers with order type \( \omega_1^{\mathrm{CK}} \) that is computable with an </nowiki>[[Infinite time Turing machine|''infinite time'' Turing machine]], since they are able to compute whether a given Turing machine computes a well-order or not,<ref>Hamkins Lewis 2000, theorems that mention "\(\mathrm{WO}\)"</ref>. Also, note that given computable well-orders on the natural numbers with order types \( \alpha \) and \( \beta \), it is possible to construct computable well-orders with order-types \( \alpha + \beta \), \( \alpha \cdot \beta \) and \( \alpha^{\beta} \) and much more, meaning that the Church-Kleene ordinal is not pathological and in fact a limit ordinal, [[Epsilon numbers|epsilon number]], [[Strongly critical ordinal|strongly critical]], and more. <nowiki>It has a variety of other convenient definitions. One of them has to do with the constructible hierarchy - \( \omega_1^{\mathrm{CK}} \) is the least </nowiki>[[admissible]] ordinal. In other words, it is the least limit ordinal \( \alpha > \omega \) so that, for any \( \Delta_0(L_\alpha) \)-definable function \( f: L_\alpha \to L_\alpha \), then, for all \( x \in L_\alpha \), \( f<nowiki>''</nowiki>x \in L_\alpha \). That is, the set of constructible sets with rank at most \( \omega_1^{\mathrm{CK}} \) is closed under taking preimages of an infinitary analogue of the primitive recursive functions. Note that this property still holds for \( \Sigma_1(L_\alpha) \)-functions, an infinitary analogue of Turing-computable functions, which makes sense, since the ordinals below \( \omega_1^{\mathrm{CK}} \) are a very robust class and closed under computable-esque functions. It is, in particular, equivalent to the statement: for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point (equivalent to a fixed point) of \( f \) below \( \alpha \). This also is a good explanation, since it shows it's a limit of epsilon numbers (and thus itself an epsilon number), of strongly critical numbers, etc. and why it's greater than recursive ordinals like the [[Bird ordinal]]. However, the case with \( \Sigma_2(L_\alpha) \)-functions produces a much stronger notion known as \( \Sigma_2 \)-admissibility. <nowiki>Note that, like how there is a fine hierarchy of recursive ordinals and functions on them, there is a fine hierarchy of nonrecursive ordinals, above \( \omega_1^{\mathrm{CK}} \), arguably richer.</nowiki> <nowiki>One thing to note is that many known ordinal collapsing functions are, or should be, \( \Sigma_1(L_{\omega_1^{\mathrm{CK}}}) \)-definable. Thus, the countable collapse is actually a recursive collapse, and replacing \( \Omega \) with \( \omega_1^{\mathrm{CK}} \) in an ordinal collapsing function is a possibility. While many authors do this, since it allows them to use structure more efficiently and not assume </nowiki>[[Large cardinal|large cardinal axioms]], more cumbersome proofs would be necessary, and this has led many authors such as Rathjen to instead opt for the traditional options, or use uncountable intermediates between countable nonrecursive fine structure and large cardinals, such as the reducibility hierarchy. 45c38b1929e1978b08cb9847b9dba8b1035bf795 711 671 2024-03-25T16:46:55Z CreeperBomb 30 wikitext text/x-wiki <nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "</nowiki>[[Recursive ordinal|recursive]]<nowiki> ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning there must be some ordinals which are still countable but they aren't recursive - i.e: they're so large that all well-orders they code are so complex that they are uncomputable. The least such is the Church-Kleene ordinal. Note that there is still a well-order on the natural numbers with order type \( \omega_1^{\mathrm{CK}} \) that is computable with an </nowiki>[[Infinite time Turing machine|''infinite time'' Turing machine]], since they are able to compute whether a given Turing machine computes a well-order or not,<ref>Hamkins Lewis 2000, theorems that mention "\(\mathrm{WO}\)"</ref>. Also, note that given computable well-orders on the natural numbers with order types \( \alpha \) and \( \beta \), it is possible to construct computable well-orders with order-types \( \alpha + \beta \), \( \alpha \cdot \beta \) and \( \alpha^{\beta} \) and much more, meaning that the Church-Kleene ordinal is not pathological and in fact a limit ordinal, [[Epsilon numbers|epsilon number]], [[Strongly critical ordinal|strongly critical]], and more. <nowiki>It has a variety of other convenient definitions. One of them has to do with the constructible hierarchy - \( \omega_1^{\mathrm{CK}} \) is the least </nowiki>[[admissible]] ordinal. In other words, it is the least limit ordinal \( \alpha > \omega \) so that, for any \( \Delta_0(L_\alpha) \)-definable function \( f: L_\alpha \to L_\alpha \), then, for all \( x \in L_\alpha \), \( f<nowiki>''</nowiki>x \in L_\alpha \). That is, the set of constructible sets with rank at most \( \omega_1^{\mathrm{CK}} \) is closed under taking preimages of an infinitary analogue of the primitive recursive functions. Note that this property still holds for \( \Sigma_1(L_\alpha) \)-functions, an infinitary analogue of Turing-computable functions, which makes sense, since the ordinals below \( \omega_1^{\mathrm{CK}} \) are a very robust class and closed under computable-esque functions. It is, in particular, equivalent to the statement: for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point (equivalent to a fixed point) of \( f \) below \( \alpha \). This also is a good explanation, since it shows it's a limit of epsilon numbers (and thus itself an epsilon number), of strongly critical numbers, etc. and why it's greater than recursive ordinals like the [[Bird ordinal]]. However, the case with \( \Sigma_2(L_\alpha) \)-functions produces a much stronger notion known as \( \Sigma_2 \)-admissibility. <nowiki>Note that, like how there is a fine hierarchy of recursive ordinals and functions on them, there is a fine hierarchy of nonrecursive ordinals, above \( \omega_1^{\mathrm{CK}} \), arguably richer.</nowiki> <nowiki>One thing to note is that many known ordinal collapsing functions are, or should be, \( \Sigma_1(L_{\omega_1^{\mathrm{CK}}}) \)-definable. Thus, the countable collapse is actually a recursive collapse, and replacing \( \Omega \) with \( \omega_1^{\mathrm{CK}} \) in an ordinal collapsing function is a possibility. While many authors do this, since it allows them to use structure more efficiently and not assume </nowiki>[[Large cardinal|large cardinal axioms]], more cumbersome proofs would be necessary, and this has led many authors such as Rathjen to instead opt for the traditional options, or use uncountable intermediates between countable nonrecursive fine structure and large cardinals, such as the reducibility hierarchy. a74341986b31655cc0b8d3e3e2aab1658eaf344b 0 191 672 480 2024-03-25T05:54:00Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] A bijection between two sets, \(X\) and \(Y\), is a "one-to-one pairing" of their elements. Formally, it is a function \(f: X \to Y\) (which can be encoded as a subset of \(X \times Y\)) so that: * Different elements of \(X\) are sent to different elements of \(Y\). * Every element of \(Y\) has some element of \(X\) which is sent to \(Y\). The first property is known as injectivity, or being 1-1, and can be formally be written as \(f(x) = f(y)\) only if \(x = y\). The second property is known as surjectivity, or being onto, and can be formally written as, for all \(y \in Y\), there is \(x \in X\) so that \(f(x) = y\). Bijections are used to define [[Cardinal|cardinals]] and cardinality. 4f693b21f72c80c55f121da756a231f016403686 712 672 2024-03-25T16:47:09Z CreeperBomb 30 wikitext text/x-wiki A bijection between two sets, \(X\) and \(Y\), is a "one-to-one pairing" of their elements. Formally, it is a function \(f: X \to Y\) (which can be encoded as a subset of \(X \times Y\)) so that: * Different elements of \(X\) are sent to different elements of \(Y\). * Every element of \(Y\) has some element of \(X\) which is sent to \(Y\). The first property is known as injectivity, or being 1-1, and can be formally be written as \(f(x) = f(y)\) only if \(x = y\). The second property is known as surjectivity, or being onto, and can be formally written as, for all \(y \in Y\), there is \(x \in X\) so that \(f(x) = y\). Bijections are used to define [[Cardinal|cardinals]] and cardinality. a46f0bf8d324f77cbd1bfe0184d107303803ccd7 0 197 673 505 2024-03-25T05:54:08Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the [[axiom of choice]], and \(\aleph_1\) is the smallest [[Countability|uncountable]] cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the context of the [[axiom of determinacy]], it holds that, for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size, and yet \(2^{\aleph_0} \neq \aleph_1\) (in particular, the two are incomparable, since one needs choice to prove cardinals are linearly ordered). The continuum hypothesis was originally posed by Georg Cantor after his proof of [[Cantor's diagonal argument|the diagonal argument]]. Cantor was unable to prove or disprove the continuum hypothesis, primarily because he was unable to find a [[Well-ordered set|well-order]] of the reals. It is now known that there is no definable well-order on the reals: therefore, this approach can't work. Solving the continuum hypothesis was the first on Hilbert's famous list of problems to be solved in the 20th century. After proving his first incompleteness theorem, Gödel had a suspicion that the continuum hypothesis may be independent of [[ZFC]]. He proved that the continuum hypothesis could not be disproved in ZFC, by showing that it was true in his [[Inner model theory|inner model]] [[Constructible hierarchy|\(L\)]].<ref>Any set theory text</ref> (Assuming ZFC is consistent to begin with, the alternative being that ZFC proves and refutes any statement by the principle of explosion.) Cohen then proved it could not be proved in ZFC either, assuming ZFC is consistent: given a countable standard transitive model \(M\) of ZFC, he proved that there was a forcing extension \(M[G]\) which added \(\aleph_2^M\) reals, and therefore that the continuum hypothesis fails within \(M[G]\).<ref>Any text about forcing</ref> The powerful method of forcing could also be used to show the opposite: if \(M\) was a countable standard transitive model of ZFC, then there was a forcing extension \(M[G]\) that added a surjection from \(\aleph_1^M \to \mathfrak{c}^M\), and therefore the continuum hypothesis holds in \(M[G]\). As such, the continuum hypothesis is independent of ZFC, if ZFC is consistent. It is therefore often regarded as one of the biggest unsolved problems in set theory. 5e772b92fc185d611cb7375e0d4882c739fe0312 713 673 2024-03-25T16:47:23Z CreeperBomb 30 wikitext text/x-wiki The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the [[axiom of choice]], and \(\aleph_1\) is the smallest [[Countability|uncountable]] cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the context of the [[axiom of determinacy]], it holds that, for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size, and yet \(2^{\aleph_0} \neq \aleph_1\) (in particular, the two are incomparable, since one needs choice to prove cardinals are linearly ordered). The continuum hypothesis was originally posed by Georg Cantor after his proof of [[Cantor's diagonal argument|the diagonal argument]]. Cantor was unable to prove or disprove the continuum hypothesis, primarily because he was unable to find a [[Well-ordered set|well-order]] of the reals. It is now known that there is no definable well-order on the reals: therefore, this approach can't work. Solving the continuum hypothesis was the first on Hilbert's famous list of problems to be solved in the 20th century. After proving his first incompleteness theorem, Gödel had a suspicion that the continuum hypothesis may be independent of [[ZFC]]. He proved that the continuum hypothesis could not be disproved in ZFC, by showing that it was true in his [[Inner model theory|inner model]] [[Constructible hierarchy|\(L\)]].<ref>Any set theory text</ref> (Assuming ZFC is consistent to begin with, the alternative being that ZFC proves and refutes any statement by the principle of explosion.) Cohen then proved it could not be proved in ZFC either, assuming ZFC is consistent: given a countable standard transitive model \(M\) of ZFC, he proved that there was a forcing extension \(M[G]\) which added \(\aleph_2^M\) reals, and therefore that the continuum hypothesis fails within \(M[G]\).<ref>Any text about forcing</ref> The powerful method of forcing could also be used to show the opposite: if \(M\) was a countable standard transitive model of ZFC, then there was a forcing extension \(M[G]\) that added a surjection from \(\aleph_1^M \to \mathfrak{c}^M\), and therefore the continuum hypothesis holds in \(M[G]\). As such, the continuum hypothesis is independent of ZFC, if ZFC is consistent. It is therefore often regarded as one of the biggest unsolved problems in set theory. f5a65f4e884cb3c95aefbe318aa56b826ba486af 0 208 674 572 2024-03-25T05:54:27Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is the minimal weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]] cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is [[ordinal-definable]]. In fact, an even stronger theorem holds: assume the HOD hypothesis holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable. 50e26e22d064623072b1584bc2bc1694fdf9c6af 714 674 2024-03-25T16:48:01Z CreeperBomb 30 wikitext text/x-wiki Extender models are [[Inner model theory|inner models]], which have similar fine structure to [[Constructible hierarchy|Gödel's \(L\)]], but which are able to accommodate large cardinals, typically at the level of [[Measurable|measurable cardinals]] and above. Extender models are either constructed - where they have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) - and have their fine structure analysed, or are defined in a more broad scope and the abstract properties of all such models considered. For example, a notion of a (weak) extender model for [[Supercompact|supercompactness]] has been isolated, and the properties of such models analysed, however an actual construction of such a model is extremely difficult and has not yet been carried out. N.b: the [[HOD dichotomy]] implies that \(\mathrm{HOD}\) may be a weak extender model for supercompactness, but it lacks the necessary fine structure, as mentioned on [[Ordinal definable|the page]]. In general, if a large cardinal property \(\Phi(\kappa)\) is equivalent to "for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) on \(X_\xi\) so that \(\psi(U_\xi, \xi)\) holds", where \(X_\xi\) is an arbitrary set and \(\psi\) is an arbitrary formula, the assertion "\(N\) is a weak extender model for \(\Phi(\kappa)\)" means that, for all \(\xi \geq \kappa\), there is an [[Filter|ultrafilter]] \(U_\xi\) so that \(\psi(U_\xi, \xi)\) holds, \(N \cap X_\xi \in U_\xi\) and \(U_\xi \cap N \in N\). This notion is obviously designed to generalize the properties of a particular constructed extender model - for example, "\(N\) is a weak extender model for \(\kappa\)'s [[Measurable|measurability]]" means that there is an ultrafilter \(U\) witnessing \(\kappa\)'s measurability so that \(U \cap N \in N\) and if \(\kappa\) is a measurable cardinal and \(U\) witnesses this, then \(L[U]\) is the minimal weak extender model for \(\kappa\)'s measurability. "\(N\) is a weak extender model for \(\kappa\)'s supercompactness" means that, for all \(\lambda \geq \kappa\), there is an ultrafilter \(U\) witnessing \(\kappa\)'s \(\lambda\)-supercompactness so that \(N \cap [\lambda]^{< \kappa} \in U\) and \(U \cap N \in N\). Weak extender models for supercompactness not only are [[Covering property|universal]] but also possess the following potent closure property: if \(N\) is a weak extender model for \(\kappa\)'s supercompactness and \(\lambda > \kappa\) is a cardinal in \(N\), then, for any elementary embedding \(j: H(\lambda^+)^N \to H(j(\lambda)^+)^N\) with critical point at least \(\kappa\), we have \(j \in N\). This follows from the weak extender model [[Extendible|version of Magidor's lemma]]. Furthermore, If \(E\) is an \(N\)-[[extender]] with length \(\eta\) and the embedding \(j_E\) generated by \(E\) has critical point at least \(\kappa\), and, for each \(A \subseteq \eta\), we have \(j_E(A) \cap \eta \in N\), then \(E \cap N \in N\). Therefore, the [[HOD dichotomy|HOD hypothesis]] implies that, if \(\delta\) is an [[extendible]] cardinal, \(\lambda > \kappa\) is a cardinal in \(\mathrm{HOD}\), and \(j: H(\lambda^+)^{\mathrm{HOD}} \to H(j(\lambda)^+)^{\mathrm{HOD}}\) is an elementary embedding with critical point at least \(\delta\), \(j\) is [[ordinal-definable]]. In fact, an even stronger theorem holds: assume the HOD hypothesis holds and there is an [[extendible]] cardinal. Then there is an ordinal \(\lambda\) so that, for all \(\gamma > \lambda\), if \(j: \mathrm{HOD} \cap V_{\gamma+1} \to \mathrm{HOD} \cap V_{j(\gamma)+1}\) is an elementary embedding with \(j(\lambda) = \lambda\) (so the critical point is either above or far below \(\lambda\)), then \(j\) is hereditarily ordinal-definable. 6d06d071f44cd95abbcf051a9061f66c47ae7fb8 0 178 675 501 2024-03-25T05:54:33Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is trivial and intuitive. For example, one can see that the axiom of choice is equivalent to the assertion that the [[Cartesian product]] of any collection of nonempty sets is nonempty. Note that the assertion that the Cartesian product of finitely many nonempty sets is nonempty is obvious, but it's possible to define Cartesian product of infinitely many sets. Despite these simple characteristics, the axiom of choice is not a theorem of ZF, and it has some consequences that may be counterintuitive. For example, the axiom of choice is highly nonconstructive and doesn't actually tell somebody what that choice function looks like. Similarly, the axiom of choice tells us there is some well-order on the real numbers, but it is a theorem that there is no well-order on the real numbers. In general, the axiom of choice implies every set can be well-ordered: that is, for every set \(X\), there is a relation on \(X\) which imbues it with the structure necessary for it to be considered a [[well-ordered set]]. All finite sets, and even countable sets, can be trivially well-ordered, and in most cases this well-ordering will be definable, but the uncountable case is unclear. The proof that the axiom of choice implies that every set can be well-ordered is relatively simple. Namely, let \(Y\) be the family of subsets of \(X\). Let \(f\) be a choice function for \(Y\). Then define, via transfinite recursion, the [[ordinal]] indexed sequence \(a_\xi\) of elements of \(X\) by \(a_\xi = f(X \setminus \{a_\eta: \eta < \xi\})\). Every element of \(X\) shows up somewhere in this sequence. Therefore, define \(\leq\) by \(a_\xi \leq a_\eta\) iff \(\xi \leq \eta\). This is well-defined, and it is a well-order since the ordinals are well-ordered. Furthermore, the axiom of choice implies the law of excluded middle, which means constructivist mathematicians tend to work in ZF rather than ZFC. Lastly, and most famously, the axiom of choice implies the [[Banach-Tarski paradox]]. In particular, using the axiom of choice, it's possible to decompose any ball in 3D space into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. This is counterintuitive, but not truly paradoxical as the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. 0b6c4528290caad570556d87219a245830cd07e6 715 675 2024-03-25T16:48:12Z CreeperBomb 30 wikitext text/x-wiki The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of [[ZFC]]. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of [[Set|sets]], it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is trivial and intuitive. For example, one can see that the axiom of choice is equivalent to the assertion that the [[Cartesian product]] of any collection of nonempty sets is nonempty. Note that the assertion that the Cartesian product of finitely many nonempty sets is nonempty is obvious, but it's possible to define Cartesian product of infinitely many sets. Despite these simple characteristics, the axiom of choice is not a theorem of ZF, and it has some consequences that may be counterintuitive. For example, the axiom of choice is highly nonconstructive and doesn't actually tell somebody what that choice function looks like. Similarly, the axiom of choice tells us there is some well-order on the real numbers, but it is a theorem that there is no well-order on the real numbers. In general, the axiom of choice implies every set can be well-ordered: that is, for every set \(X\), there is a relation on \(X\) which imbues it with the structure necessary for it to be considered a [[well-ordered set]]. All finite sets, and even countable sets, can be trivially well-ordered, and in most cases this well-ordering will be definable, but the uncountable case is unclear. The proof that the axiom of choice implies that every set can be well-ordered is relatively simple. Namely, let \(Y\) be the family of subsets of \(X\). Let \(f\) be a choice function for \(Y\). Then define, via transfinite recursion, the [[ordinal]] indexed sequence \(a_\xi\) of elements of \(X\) by \(a_\xi = f(X \setminus \{a_\eta: \eta < \xi\})\). Every element of \(X\) shows up somewhere in this sequence. Therefore, define \(\leq\) by \(a_\xi \leq a_\eta\) iff \(\xi \leq \eta\). This is well-defined, and it is a well-order since the ordinals are well-ordered. Furthermore, the axiom of choice implies the law of excluded middle, which means constructivist mathematicians tend to work in ZF rather than ZFC. Lastly, and most famously, the axiom of choice implies the [[Banach-Tarski paradox]]. In particular, using the axiom of choice, it's possible to decompose any ball in 3D space into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. This is counterintuitive, but not truly paradoxical as the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. 8d6ed23c0710e26a925dea895d68bfecd3bc2c16 0 161 676 441 2024-03-25T05:54:41Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any nonempty \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity: For all \(a\), \(a \leq a\). These properties were intended to idealize and generalize the properties of the ordering on the natural numbers. Note that, under their usual ordering, the natural numbers are well-ordered, but the integers, nonnegative rationals, and any real number interval are not. The [[axiom of choice]] implies, however, that any set can be well-ordered, just the prior examples aren't well-ordered with respect to their natural ordering. As mentioned just now, the axiom of choice implies there is a well-order of the reals, but it is not definable, while the [[axiom of determinacy]] implies that there is no well-order on the reals whatsoever. Well-ordered sets are part of the motivation of [[Ordinal|ordinals]], and are used to define their equivalence class definition. In particular, for any well-ordered set \(X\), there is an ordinal \(\alpha\) and a map \(\pi: X \to \alpha\) so that \(x \leq y\) implies \(\pi(x) \leq \pi(y)\): the unique such ordinal is called its order-type, and such a function \(\pi\) is called an order-isomorphism. The order-type of the natural numbers is [[Omega|\(\omega\)]], where the map \(\pi\) is just the identity function, and any ordinal is its own order-type. Also, any [[countable]] set has a countable order-type, and any [[finite]] set has a finite order-type (which is necessarily equal to its [[cardinality]], unlike the case with infinite sets). You can see that, assuming the axiom of choice, the well-foundedness criterion is equivalent to there not being an infinite descending chain. Assume \(S \subseteq X\), and there is no infinite descending chain. Assume towards contradiction that \(S\) has no minimal element. Use AC to define a choice function \(f\) for \(\{X: X \subseteq S\}\). Let \(s_0 = f(S) \in S\), and then \(s_{n+1} = f(\{x \in S: x \leq s_n \land x \neq s_n\})\). \(\{x \in S: x \leq s_n \land x \neq s_n\}\) is always nonempty, because else \(s_n\) would be a minimal element of \(S\). Then \(s\) forms an infinitely descending chain. Contradiction! For the converse, assume that every nonempty subset has a minimal element, and assume \((s_i)_{i = 0}^\infty\) is an infinite sequence with \(s_i \leq s_j\) for \(j < i\). Let \(S = \{s_i: i \in \mathbb{N}\}\). Therefore \(s\) is eventually constant, and so can't be infinitely ''strictly'' decreasing. 62971118fbc16715dbf8b5be97b0aed754308538 716 676 2024-03-25T16:48:24Z CreeperBomb 30 wikitext text/x-wiki A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any nonempty \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity: For all \(a\), \(a \leq a\). These properties were intended to idealize and generalize the properties of the ordering on the natural numbers. Note that, under their usual ordering, the natural numbers are well-ordered, but the integers, nonnegative rationals, and any real number interval are not. The [[axiom of choice]] implies, however, that any set can be well-ordered, just the prior examples aren't well-ordered with respect to their natural ordering. As mentioned just now, the axiom of choice implies there is a well-order of the reals, but it is not definable, while the [[axiom of determinacy]] implies that there is no well-order on the reals whatsoever. Well-ordered sets are part of the motivation of [[Ordinal|ordinals]], and are used to define their equivalence class definition. In particular, for any well-ordered set \(X\), there is an ordinal \(\alpha\) and a map \(\pi: X \to \alpha\) so that \(x \leq y\) implies \(\pi(x) \leq \pi(y)\): the unique such ordinal is called its order-type, and such a function \(\pi\) is called an order-isomorphism. The order-type of the natural numbers is [[Omega|\(\omega\)]], where the map \(\pi\) is just the identity function, and any ordinal is its own order-type. Also, any [[countable]] set has a countable order-type, and any [[finite]] set has a finite order-type (which is necessarily equal to its [[cardinality]], unlike the case with infinite sets). You can see that, assuming the axiom of choice, the well-foundedness criterion is equivalent to there not being an infinite descending chain. Assume \(S \subseteq X\), and there is no infinite descending chain. Assume towards contradiction that \(S\) has no minimal element. Use AC to define a choice function \(f\) for \(\{X: X \subseteq S\}\). Let \(s_0 = f(S) \in S\), and then \(s_{n+1} = f(\{x \in S: x \leq s_n \land x \neq s_n\})\). \(\{x \in S: x \leq s_n \land x \neq s_n\}\) is always nonempty, because else \(s_n\) would be a minimal element of \(S\). Then \(s\) forms an infinitely descending chain. Contradiction! For the converse, assume that every nonempty subset has a minimal element, and assume \((s_i)_{i = 0}^\infty\) is an infinite sequence with \(s_i \leq s_j\) for \(j < i\). Let \(S = \{s_i: i \in \mathbb{N}\}\). Therefore \(s\) is eventually constant, and so can't be infinitely ''strictly'' decreasing. 640897c2045ec1d7f3d70efa842f8bb46cef8029 0 199 677 504 2024-03-25T05:54:59Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The Banach-Tarski is a famous, counterintuitive consequence of the [[axiom of choice]]. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. The proof requires the axiom of choice, and, therefore, the truth of the Banach-Tarski paradox is a common argument against the usage of the axiom of choice. a379b64057cb4169dbe7b4c32852f4342fa971bf 0 9 678 489 2024-03-25T05:55:31Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. The existence of \(\omega\) is guaranteed by the [[axiom of infinity]]. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is considered by some to be the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. b4965a739af6f2e716e70f9b2e788671fa66668f 0 168 679 493 2024-03-25T05:55:41Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Cardinals (or cardinal numbers) are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. In general, one may informally describe cardinals as numbers, [[finite]] or [[infinite]], which are meant to describe how many objects there are in a collection. The cardinality of a [[set]] is the (unique) cardinal representing its size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the [[axiom of choice]], since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> Here, the cardinality of a set is just the unique equivalence class which the set belongs to. However, in the context of the axiom of choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). The cardinality of a set is then defined as the minimal ordinal which it bijects with, although it might require some effort to show that the cardinality of a set is always an initial ordinal. Using the [[Ordinal#Von Neumann definition|von Neumann]] interpretation of ordinals, and the initial ordinal interpretation of cardinals, one gets that any [[Natural numbers|natural number]] is a cardinal. In particular, under the initial ordinal definition, a cardinal is a natural number if and only if it is finite. Under the equivalence class definition, a cardinal is a natural number if and only if some (furthermore, any) element of it is finite. Again in the initial ordinal definition, every cardinal is an ordinal yet there are many (infinite) ordinals which are not cardinals. However, under the equivalence class definition, no cardinal is an ordinal. Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is [[Aleph 0|\(\aleph_0\)]] - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. 5902962e2d400ccb8c8db6d5c71ad486ad3aee7a 0 175 680 440 2024-03-25T05:56:29Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Lambda calculus is a simple system of computation introduced by Alonzo Church. in which functions, and the operations of abstraction and application, act as primitive operations and objects. [[Natural numbers]] can be encoded in the lambda calculus using a system known as Church numerals. It's been proven that lambda calculus and Turing machines are able to compute the same processes, which led to the independently formulated Church-Turing thesis that all Turing-complete methods of computation are equivalent. John Tromp has invented the system of binary lambda calculus, an extremely compact and efficient encoding of lambda calculus, which he used to define a variant of [[Kleene's O]], and used it to calculate the fundamental sequence of [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]]. e4f560ace298d9d510d08f73456cf3d6d635084a 0 59 681 169 2024-03-25T05:56:53Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences of other such objects, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. Notable ordinal notation systems include: - [[Cantor normal form]] - [[Primitive sequence system]] - [[Pair sequence system]] - Ordinal notation systems associated to [[ordinal collapsing functions]] - [[Taranovsky's ordinal notations]] (the ones that are well-ordered) - [[Patterns of resemblance]] - [[Bashicu matrix system]] - [[Y sequence]] (as long as it is well-ordered) As can be seen in this list, proposed ordinal notation systems need to be [[Proving well-orderedness | proven]] well-ordered in order to be considered ordinal notation systems with certainty, and there are notable cases where this is not proven yet. c20415b189c855987ae155cdef709fd4a96a1c62 0 86 682 298 2024-03-25T05:57:01Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). Buchholz further extended this to [[Buchholz's psi-functions|his famous set]] of collapsing functions, whose limit is the much larger [[Buchholz ordinal]]. 316017dd368a22b73f8baa537d041418af2b87ab 0 48 683 200 2024-03-25T05:57:08Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]]<ref>D. Probst, [https://boris.unibe.ch/108693/1/pro17.pdf#page=153 A modular ordinal analysis of metapredicative subsystems of second-order arithmetic] (2017), p.153</ref> or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 1503cd6e41d14312fd605892b3df4b8b205cd13f 0 223 684 596 2024-03-25T05:57:18Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>[https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> 89da6e9cd6a371d5894467991cb31a42c82d2d20 0 174 685 439 2024-03-25T05:58:14Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> The second-order extension of Peano arithmetic is [[second-order arithmetic]], a significantly more expressive system. One subsystem, \(\mathrm{ACA}_0\) (arithmetical comprehension axiom) is first-order conservative over Peano arithmetic, and is first-order categorical: that is, the first-order parts of any two models of \(\mathrm{ACA}_0\) are isomorphic.<ref>Was Sind und was Sollen Die Zahlen?, Dedekind, R., ''Cambridge Library Collection - Mathematics'', Cambridge University Press</ref> However, Peano arithmetic itself is not categorical, and has many nonstandard models. The set of finite von Neumann [[Ordinal|ordinals]], paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto a \cup \{a\}\) is a model of Peano arithmetic, and one of the most "natural" models of Peano arithmetic. Alternatively, the set of Zermelo ordinals, paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto \{a\}\) is a model of Peano arithmetic. However, the natural functions and relations \(+\), \(\cdot\) and \(<\) in this structure are more complex to describe. Peano arithmetic, minus the axiom schema of induction and plus the axiom \(\forall y (y = 0 \lor \exists x (S(x) = y))\) (which is a theorem of Peano arithmetic but requires induction), is known as Robinson arithmetic, and has proof-theoretic ordinal [[Omega|\(\omega\)]]. As mentioned previously, the axiom schema of induction gives Peano arithmetic a majority of its strength, which is shown by the fact that it has proof-theoretic ordinal [[Epsilon numbers|\(\varepsilon_0\)]], famously shown by Gentzen. Similarly, Robinson arithmetic is unable to show the function \(f_\omega\) in the fast-growing hierarchy is total, while the least rank of the fast-growing hierarchy which outgrows all computable functions provably total in Peano arithmetic is \(f_{\varepsilon_0}\). e3a2074f8d52170176e4f2e36983124cd1ebcba2 0 166 686 396 2024-03-25T05:58:24Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The axiom of infinity is a common mathematical axiom included in theories such as [[Kripke-Platek set theory]] or [[ZFC]]. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that [[Omega|\(\omega\)]] exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define [[Ordinal|ordinals]]. For example, \(V_\omega\), the set of [[Hereditarily finite set|hereditarily finite sets]], is a model of [[ZFC]] minus the axiom of infinity. 28f297e8d269a0463de97663c8766f03b63a8c55 0 165 687 395 2024-03-25T05:58:43Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper class, and the [[Burali–Forti paradox]] shows that \(\mathrm{Ord}\). Also, all inner models, such as \(L\), are proper classes. Under the equivalence class definition of [[natural numbers]], [[Ordinal|ordinals]] and [[Cardinal|cardinals]], all numbers (other than [[0]]) are proper classes, which is one of the downsides of this method. [[ZFC]] is a strictly first-order theory, and thus a true treatment of proper classes is not possible within it. Instead, one only uses definable classes such as \(V\), \(\mathrm{Ord}\), and uses \(x \in X\) as a shorthand for \(\varphi(x)\), where \(\varphi\) is a first-order theory. Working within a true second-order treatment of proper, including non-definable, classes is a very powerful tool and can prove, for example, the existence of a proper class of worldly cardinals.<ref>[http://web.archive.org/web/20200916182741/https://philippschlicht.github.io/meetings/files/secondOrderSetTheoryBristol.pdf]</ref> af145b1041994c4c8ae607b976e639737ab5a9ab 0 96 688 414 2024-03-25T05:59:05Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's [[Buchholz's psi-functions|original set]] of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). Connection to Buchholz hydras 0371a28b52bcab4c041b7f4ca7607a0380ab6606 0 177 689 609 2024-03-25T05:59:15Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. Particularly, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, the cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. bb5d073dc5f590f090a81e74adc21deed5aae4bf 0 202 690 657 2024-03-25T05:59:34Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The cofinality of an [[ordinal]] \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's [[countable]], then it's exactly \(\omega\). It is easy to see that \(\mathrm{cof}(\alpha) \leq \alpha\) for all \(\alpha\), because the identity has unbounded range. Also, \(\mathrm{cof}(\mathrm{cof}(\alpha)) = \mathrm{cof}(\alpha)\), because if there is a \(\delta < \mathrm{cof}(\alpha)\) and maps \(f: \delta \to \mathrm{cof}(\alpha)\), \(g: \mathrm{cof}(\alpha) \to \alpha\) with unbounded range, then \(g \circ f: \delta \to \alpha\) also has unbounded range, contradicting minimality of \(\mathrm{cof}(\alpha)\). An ordinal is regular if it is equal to its own cofinality, else it is singular. So: * [[0]], [[1]] and [[Omega|\(\omega\)]] are regular. * All natural numbers other than \(0\) and \(1\) are singular. * All countable infinite ordinals other than \(\omega\) are singular. * \(\mathrm{cof}(\alpha)\) is regular for any \(\alpha\). Cofinality is used in the definition of [[Inaccessible cardinal|weakly inaccessible]] cardinals. ==Without choice== Citation about every uncountable cardinal being singular being consistent with ZF 95008a1fee2d1bab9900a9af516110cbc6bda180 0 226 691 615 2024-03-25T05:59:40Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Nothing OCF is a weak [[Ordinal collapsing function|OCF]], defined by CatIsFluffy. It is similar to many other OCFs in definition, but omits addition. Therefore, the growth rate is much, much slower. It is believed to correspond to a weak version of [[Extended Buchholz's function]], also defined by omitting addition, and that it catches up to the ordinary version of EBOCF by [[Extended Buchholz ordinal|EBO]]. However, no proof of either of these claims has been given and they remain open questions. 6fbca549cea2d8b3f84933ed25b2eb9deb0eabfe 0 163 692 427 2024-03-25T05:59:52Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it with the Hebrew later for Tav and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for [[Reflection principle|reflection principles]], with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the [[Burali–Forti paradox]]), so large it almost "transcended" itself, and associated it metaphysically with God. Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology. ==As justification for reflection== Later authors have connected Cantor's remark that absolute infinity "can not be conceived" to reflection principles. For example, Maddy states:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic, vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces ''reflection'' to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \( V \) is already true of some [\( V_\alpha \)]. 65ca2a44f6cfb961ec39f1f4c6f1bd21a1051860 0 114 693 652 2024-03-25T06:00:03Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additive principal ordinal is 1 since \(0 + 0 < 1\), and all additive principal ordinals other than 1 are limit ordinals. In particular, as can be seen from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation), additive principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some ordinal \(\gamma\). As such, the second infinite additive principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additive principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of the limits of additive principal ordinals is \(\omega^{\omega^2}\). Additive principal ordinals can be generalized to multiplicative principal ordinals and exponential principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicative principal ordinals are to additive principal ordinals as additive principal ordinals are to limit ordinals. However, exponential principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the [[epsilon numbers]]. 2aa9fa58f24029212f3a1c311fc96ea452570650 0 192 694 497 2024-03-25T06:01:40Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. [[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The question of whether \(\omega_1\), the least uncountable [[cardinal]], and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the [[continuum hypothesis]]. Surprisingly, assuming its consistency, this is neither provable nor disprovable in [[ZFC]]! The existence of an arbitrary set's powerset is not provable from [[Kripke-Platek set theory|KP]], even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC. 29ac361c185a1ad7fcf5529e6a05d09755719e23 0 198 695 650 2024-03-25T06:01:51Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Hilbert's Grand Hotel is a famous analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the often unintuitive ways infinite [[bijections]] work, it is actually possible to still fit many more people. Firstly, if there is a single new guest who wants a room, it is possible to accommodate by simply telling everyone to move up one room - so the person checked in Room 0 moves to Room 1, the person checked in Room 1 moves to Room 2, and so on. Because every room has a room coming after it, everybody who was checked in still has a room. Yet Room 0 is now empty - the new guest can check in there. This is analogous to the proof that [[Omega|\(\omega\)]] and \(\omega+1\) are equinumerous (that is, they have the same [[cardinality]]). Similarly, if someone in room \(n\) checks out of the hotel, then everybody in room \(m\) for \(m > n\) can move one room to the left, and all the rooms will be filled again. One can also accommodate countably infinitely many new guests, by requiring that every current guest in Room \(n\) goes to Room \(2n\) and that the \(n\)th new guest go to Room \(2n+1\). The first part frees up all the odd-numbered rooms, which the new guests can fill up. Therefore, \(\omega 2\) is equinumerous with \(\omega\). In fact, it's even possible to accommodate a countably infinite collection of countably infinitely many sets of new guests! One can assign the current guest in room \(n\) to room \(2^n\), the \(n\)th guest in the first collection of new guests to room \(3^n\), the \(n\)th guest in the next collection of new guests to room \(5^n\), then \(7^n\), \(11^n\), and so on. Because there are infinitely many prime numbers, and powers of primes never overlap, everybody can be accommodated - even with many rooms now empty, such as room 6, which isn't a power of any prime number! However, not every infinite batch of guests can fit in Hilbert's Grand Hotel. If a bus brings infinitely many guests whose names are all infinite strings made up of "a" and "b", and every string has a guest, not all of the guests can fit. In fact, it's possible to pair up each name to a real number, showing that there are more real numbers than natural numbers, even though there are infinitely many of both! 1836c20584bc38b225e724c8be0e7a88a4ac9382 0 54 696 617 2024-03-25T06:02:00Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu: \textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_\mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. == Extension == This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_\mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. == References == acf26ca81b27e4b16f694c671c7d41163e6a5736 0 49 697 277 2024-03-25T06:02:16Z Cobsonwabag 32 wikitext text/x-wiki <div style="position:fixed;left:0;top:0"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 0px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 400px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 300px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 800px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 0px; left: 1200px; position: fixed; float: left;"> [[File:Cobson.png|link=]] </div> <div style="top: 300px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 600px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 900px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] </div> <div style="top: 1200px; left: 1200px; position: fixed; float: left;"> [[File:coinslot.png|link=]] The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes). a2b295e6b4bc89d1a05161920b72ca78dd4c9253 0 1 699 659 2024-03-25T16:42:32Z CreeperBomb 30 wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> e2b43ff20f800c007ae08aa90b03c1ad7f9c7348 0 57 700 660 2024-03-25T16:42:58Z CreeperBomb 30 wikitext text/x-wiki A set \(M\) is admissible if \((M,\in)\) is a model of [[Kripke-Platek set theory]]. An ordinal \(\alpha\) is admissible if there exists an admissible set \(M\) such that \(M\cap\textrm{Ord}=\alpha\). This definition of admissibility is equivalent to \(L_\alpha\vDash\textrm{KP}\), which is itself equivalent to \(\alpha > \omega\), \(\alpha\) being a limit ordinal and \(L_\alpha\) being closed under preimages of \(\alpha\)-recursively enumerable functions.<ref>Admissible Sets and Structures, Barwise, J., ''Perspectives in Logic'', Cambridge University Press.</ref> The least admissible ordinal is [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]], although some authors omit the axiom of infinity from KP and consider [[Omega|\(\omega\)]] to be admissible. cbd07960a679b393b6c6c6ac52143720cf653095 0 209 701 661 2024-03-25T16:43:20Z CreeperBomb 30 wikitext text/x-wiki Extendible cardinals are a powerful [[large cardinal]] notion, which can be considered a significant strengthening of [[Weakly compact cardinal|weakly compact cardinals]], or a combination of [[supercompact]] and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible for all \(\eta > 0\). This is equivalent to a form of second-order infinitary compactness, or one of the two following other elementary embedding characterisations: * For each \(\lambda > \kappa\), there is an elementary embedding \(j: V_{\lambda+1} \to V_{j(\lambda)+1}\) with critical point \(\kappa\) so that \(j(\kappa) > \lambda\). * For each \(\lambda \geq \kappa\), there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and \(V_{j(\kappa)} \subseteq M\), and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) and \(j(\kappa) > \lambda\). While the latter clearly characterization connects extendibles to supercompacts, the same can not be said for the first. However, this follows from Magidor's lemma: a cardinal \(\kappa\) is supercompact iff, for all \(\lambda > \kappa\), there exist \(\bar{\kappa} < \bar{\lambda} < \kappa\) and an elementary embedding \(j: V_{\bar{\lambda}+1} \to V_{\lambda+1}\) with critical point \(\bar{\kappa}\) so that \(j(\bar{\kappa}) = \kappa\). If \(N\) is a [[Extender model|weak extender model]]<nowiki> for \(\kappa\)'s supercompactness, then, for all \(a \in V_\lambda\), the above characterisation of supercompactness holds and, for some \(\bar{a} \in V_{\bar{\lambda}}\):</nowiki> * \(j(\bar{a}) = a\). * <nowiki>\(j(N \cap V_{\bar{\lambda}}) = N \cap V_{\bar{\lambda}}\).</nowiki> * <nowiki>\(j \upharpoonright (N \cap V_{\bar{\lambda}}) \in N\).</nowiki> e6d689183376f08b2f6a52b86421fefbdf44e82c 0 164 702 662 2024-03-25T16:43:59Z CreeperBomb 30 wikitext text/x-wiki An [[ordinal]] is called a successor if it is equal to \(\alpha + 1\) for some other \(\alpha\). An ordinal \(\alpha\) is a successor if and only if \(\max \alpha\) exists. The smallest successor ordinal is [[1]], which is also the only successor ordinal to also be [[Additive principal ordinals|additively principal]]. The least ordinal that is not a successor, other than [[0]], is [[Omega|\(\omega\)]]. If \(\beta\) is successor, then \(\alpha+\beta\) is also successor for all \(\alpha\). However, multiplication and exponentiation do not have this property. 1343d50e2871a63796b7a70b809dbba586aa1ca6 0 46 703 663 2024-03-25T16:44:07Z CreeperBomb 30 wikitext text/x-wiki The number '''zero''', denoted 0, is the smallest natural number. It is the additive identity for natural numbers, meaning that \(a+0=0+a=a\) for any \(a\in\mathbb N\). It is also the multiplicative annihilator for natural numbers, meaning that \(a\cdot 0=0\cdot a=0\) for any \(a\in\mathbb N\). It is the number immediately before [[1]]. As with any other natural number, it may be identified with either an [[ordinal]] or [[cardinal]]. In both cases, it still satisfies the aforementioned properties, being the smallest element of the structure, an additive identity, and the multiplicative annihilator. In particular, it is equal to the identity in the monoid of [[natural numbers]] under addition. Zero is both the [[cardinality]] and the [[order type]] of the [[empty set]] \(\varnothing\). 20bc5affde1c5c0cdbaaf23697c1aeeaf6bb14f5 0 172 704 664 2024-03-25T16:44:19Z CreeperBomb 30 wikitext text/x-wiki Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''' (Main System with Passthrough), is known to be ill-founded.<ref>Discord message in #taranovsky-notations</ref> == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> == Sources == 5df3fdf76996125f2a48a5e16e4d98cf48f3078c 0 15 705 665 2024-03-25T16:44:47Z CreeperBomb 30 wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by BashicuHyudora. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of ordered pairs). Then if we let \( m_0 \) be minimal such that the last element of the sequence in \( A \) has an \( m_0 \)-parent, \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from its \( m_0 \)-parent to the element right before the last element, and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The previous copy of \( C \) if \( C \) is the \( m_0 \)-parent of the removed element and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> d63e6bec91915b916276dcdd3eed71afd1c517cf 0 199 717 677 2024-03-25T16:48:49Z CreeperBomb 30 Undo revision [[Special:Diff/677|677]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The Banach-Tarski is a famous, counterintuitive consequence of the [[axiom of choice]]. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. The proof requires the axiom of choice, and, therefore, the truth of the Banach-Tarski paradox is a common argument against the usage of the axiom of choice. f7179f8eba8bc50de9fafe9b845ba410686dc8fe 0 9 718 678 2024-03-25T16:48:59Z CreeperBomb 30 Undo revision [[Special:Diff/678|678]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki {{DISPLAYTITLE:\(\omega\)}} The [[ordinal]] '''omega''', written \(\omega\), is defined as the [[order type]] of the natural numbers \(\mathbb N\). As a [[von Neumann ordinal]], it corresponds to the naturals themselves. Note that \(\omega\) is not to be confused with [[Uncountable|\(\Omega\)]], a common notation for a much larger ordinal. The existence of \(\omega\) is guaranteed by the [[axiom of infinity]]. ==Properties== * It is the first [[infinite]] ordinal. * It is the first [[limit ordinal]]. * It is considered by some to be the first [[admissible ordinal]]. * Using the [[von Neumann cardinal assignment]], it is equal to [[aleph 0|\(\aleph_0\)]]. * It is the smallest ordinal \(\alpha\) such that \(1+\alpha=\alpha\). Every ordinal larger than it has this same property. * It is the next ordinal after [[0]] that isn't a [[successor ordinal]]. * It is additively, multiplicatively, and exponentially [[principal]]. 754bfa6b783ce83ef13394d4095358ea1b8f808f 0 168 719 679 2024-03-25T16:49:12Z CreeperBomb 30 Undo revision [[Special:Diff/679|679]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Cardinals (or cardinal numbers) are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size. In general, one may informally describe cardinals as numbers, [[finite]] or [[infinite]], which are meant to describe how many objects there are in a collection. The cardinality of a [[set]] is the (unique) cardinal representing its size. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the [[axiom of choice]], since not all sets are necessarily well-orderable.<ref>Hazewinkel, Michiel (2001). ''Encyclopaedia of Mathematics: Supplement''. Berlin: Springer. p. 458. ISBN <bdi>1-4020-0198-3</bdi>.</ref> Here, the cardinality of a set is just the unique equivalence class which the set belongs to. However, in the context of the axiom of choice, the former is more common because the objects we work with are [[Set|sets]] rather than [[Proper class|proper classes]]. In particular, a cardinal is just defined as an [[ordinal]] which does not biject with any smaller ordinal. All [[finite]] ordinals are cardinals, as well as [[Omega|\(\omega\)]]. The next cardinal after \(\omega\) is [[Uncountable|\(\omega_1\)]], aka \(\Omega\). The cardinality of a set is then defined as the minimal ordinal which it bijects with, although it might require some effort to show that the cardinality of a set is always an initial ordinal. Using the [[Ordinal#Von Neumann definition|von Neumann]] interpretation of ordinals, and the initial ordinal interpretation of cardinals, one gets that any [[Natural numbers|natural number]] is a cardinal. In particular, under the initial ordinal definition, a cardinal is a natural number if and only if it is finite. Under the equivalence class definition, a cardinal is a natural number if and only if some (furthermore, any) element of it is finite. Again in the initial ordinal definition, every cardinal is an ordinal yet there are many (infinite) ordinals which are not cardinals. However, under the equivalence class definition, no cardinal is an ordinal. Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is [[Aleph 0|\(\aleph_0\)]] - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\). If choice doesn't outright fail, one may talk about [[Well-ordered set|well-ordered]] and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]]. d366dbb4ac9ab84960c3842334c0b5eb5f7b5ed9 0 175 720 680 2024-03-25T16:49:37Z CreeperBomb 30 Undo revision [[Special:Diff/680|680]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Lambda calculus is a simple system of computation introduced by Alonzo Church. in which functions, and the operations of abstraction and application, act as primitive operations and objects. [[Natural numbers]] can be encoded in the lambda calculus using a system known as Church numerals. It's been proven that lambda calculus and Turing machines are able to compute the same processes, which led to the independently formulated Church-Turing thesis that all Turing-complete methods of computation are equivalent. John Tromp has invented the system of binary lambda calculus, an extremely compact and efficient encoding of lambda calculus, which he used to define a variant of [[Kleene's O]], and used it to calculate the fundamental sequence of [[Church-Kleene ordinal|<nowiki>\(\omega_1^{\mathrm{CK}}\)</nowiki>]]. 8836d26b70669fe47292ceaa7a76d78cef0d2779 0 59 721 681 2024-03-25T16:49:53Z CreeperBomb 30 Undo revision [[Special:Diff/681|681]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences of other such objects, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is included in the definition of proof-theoretic ordinals that uses ordinal notation systems. Notable ordinal notation systems include: - [[Cantor normal form]] - [[Primitive sequence system]] - [[Pair sequence system]] - Ordinal notation systems associated to [[ordinal collapsing functions]] - [[Taranovsky's ordinal notations]] (the ones that are well-ordered) - [[Patterns of resemblance]] - [[Bashicu matrix system]] - [[Y sequence]] (as long as it is well-ordered) As can be seen in this list, proposed ordinal notation systems need to be [[Proving well-orderedness | proven]] well-ordered in order to be considered ordinal notation systems with certainty, and there are notable cases where this is not proven yet. d688e7f18ffcf42b14b08b12b534cd6445d8a864 0 86 722 682 2024-03-25T16:50:05Z CreeperBomb 30 Undo revision [[Special:Diff/682|682]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). Buchholz further extended this to [[Buchholz's psi-functions|his famous set]] of collapsing functions, whose limit is the much larger [[Buchholz ordinal]]. 349c33822e78417ea29a95a9114eef8627651952 0 48 723 683 2024-03-25T16:50:32Z CreeperBomb 30 Undo revision [[Special:Diff/683|683]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki An '''ordinal function''' refers to a function from [[ordinal]]s to ordinals. More rarely, they refer to functions from an initial segment of the ordinals to another. Important examples include [[continuous function]]s and [[normal function]]s. Technically speaking and within [[ZF]], since [[Burali–Forti paradox|ordinals don't form a set]], one can't formally talk about functions \(f:\text{On}\to\text{On}\). However, as with the method of using Grothendieck universes, replacing \(\text{On}\) with the set of ordinals below a large enough ordinal, such as an [[inaccessible ordinal]] or even an [[uncountable]]<ref>D. Probst, [https://boris.unibe.ch/108693/1/pro17.pdf#page=153 A modular ordinal analysis of metapredicative subsystems of second-order arithmetic] (2017), p.153</ref> or [[principal]] ordinal, depending on context, is almost always enough to formally recover any results on them. As such, we still refer to them as functions from ordinals to ordinals in the wiki. 48675dad9ddcd27ed39030602b9c3127c061c2e1 0 223 724 684 2024-03-25T16:50:55Z CreeperBomb 30 Undo revision [[Special:Diff/684|684]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>[https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> bf93f321fc5b993c8f9d969d1a26ca761b4d7919 0 174 725 685 2024-03-25T16:51:05Z CreeperBomb 30 Undo revision [[Special:Diff/685|685]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> The second-order extension of Peano arithmetic is [[second-order arithmetic]], a significantly more expressive system. One subsystem, \(\mathrm{ACA}_0\) (arithmetical comprehension axiom) is first-order conservative over Peano arithmetic, and is first-order categorical: that is, the first-order parts of any two models of \(\mathrm{ACA}_0\) are isomorphic.<ref>Was Sind und was Sollen Die Zahlen?, Dedekind, R., ''Cambridge Library Collection - Mathematics'', Cambridge University Press</ref> However, Peano arithmetic itself is not categorical, and has many nonstandard models. The set of finite von Neumann [[Ordinal|ordinals]], paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto a \cup \{a\}\) is a model of Peano arithmetic, and one of the most "natural" models of Peano arithmetic. Alternatively, the set of Zermelo ordinals, paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto \{a\}\) is a model of Peano arithmetic. However, the natural functions and relations \(+\), \(\cdot\) and \(<\) in this structure are more complex to describe. Peano arithmetic, minus the axiom schema of induction and plus the axiom \(\forall y (y = 0 \lor \exists x (S(x) = y))\) (which is a theorem of Peano arithmetic but requires induction), is known as Robinson arithmetic, and has proof-theoretic ordinal [[Omega|\(\omega\)]]. As mentioned previously, the axiom schema of induction gives Peano arithmetic a majority of its strength, which is shown by the fact that it has proof-theoretic ordinal [[Epsilon numbers|\(\varepsilon_0\)]], famously shown by Gentzen. Similarly, Robinson arithmetic is unable to show the function \(f_\omega\) in the fast-growing hierarchy is total, while the least rank of the fast-growing hierarchy which outgrows all computable functions provably total in Peano arithmetic is \(f_{\varepsilon_0}\). 79ad5425e1ae702446152908fcfc6c390d646dbc 0 166 726 686 2024-03-25T16:51:30Z CreeperBomb 30 Undo revision [[Special:Diff/686|686]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The axiom of infinity is a common mathematical axiom included in theories such as [[Kripke-Platek set theory]] or [[ZFC]]. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that [[Omega|\(\omega\)]] exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define [[Ordinal|ordinals]]. For example, \(V_\omega\), the set of [[Hereditarily finite set|hereditarily finite sets]], is a model of [[ZFC]] minus the axiom of infinity. 13fbd04a7c0c4a6898a37300013ddea6575590ff 0 165 727 687 2024-03-25T16:51:50Z CreeperBomb 30 Undo revision [[Special:Diff/687|687]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper class, and the [[Burali–Forti paradox]] shows that \(\mathrm{Ord}\). Also, all inner models, such as \(L\), are proper classes. Under the equivalence class definition of [[natural numbers]], [[Ordinal|ordinals]] and [[Cardinal|cardinals]], all numbers (other than [[0]]) are proper classes, which is one of the downsides of this method. [[ZFC]] is a strictly first-order theory, and thus a true treatment of proper classes is not possible within it. Instead, one only uses definable classes such as \(V\), \(\mathrm{Ord}\), and uses \(x \in X\) as a shorthand for \(\varphi(x)\), where \(\varphi\) is a first-order theory. Working within a true second-order treatment of proper, including non-definable, classes is a very powerful tool and can prove, for example, the existence of a proper class of worldly cardinals.<ref>[http://web.archive.org/web/20200916182741/https://philippschlicht.github.io/meetings/files/secondOrderSetTheoryBristol.pdf]</ref> 6e6711095a167ebe290d07242f978aff8a90b332 0 96 728 688 2024-03-25T16:52:05Z CreeperBomb 30 Undo revision [[Special:Diff/688|688]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's [[Buchholz's psi-functions|original set]] of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the [[Buchholz ordinal]], although one could comparatively describe the difference in size as like that between [[Epsilon numbers|\( \varepsilon_0 \)]] and the [[Bachmann-Howard ordinal]]. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths). Connection to Buchholz hydras c80fdebca947ea58be4aaa744583a9b093008b1e 0 177 729 689 2024-03-25T16:52:30Z CreeperBomb 30 Undo revision [[Special:Diff/689|689]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). Disjoint union is used to define sum of cardinals, which disagrees with [[Ordinal#Ordinal arithmetic|ordinal]] sum. Particularly, the sum of two cardinalities is equal to the cardinality of their disjoint sum. It turns out that cardinal arithmetic other than cardinal exponentiation, which is highly nontrivial, is very trivial. Namely, the cardinality of the disjoint union of two sets is equal to the sum of their cardinalities, if both are finite, and else the maximum. In particular, \(X\) is infinite if and only if it is equinumerous with the disjoint union of it and itself, which could be taken to be equal to \(\{0,1\} \times X\), or something similar. d2f9119bbbc08d58971188878ff1808158a0abe0 0 202 730 690 2024-03-25T16:52:46Z CreeperBomb 30 Undo revision [[Special:Diff/690|690]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The cofinality of an [[ordinal]] \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's [[countable]], then it's exactly \(\omega\). It is easy to see that \(\mathrm{cof}(\alpha) \leq \alpha\) for all \(\alpha\), because the identity has unbounded range. Also, \(\mathrm{cof}(\mathrm{cof}(\alpha)) = \mathrm{cof}(\alpha)\), because if there is a \(\delta < \mathrm{cof}(\alpha)\) and maps \(f: \delta \to \mathrm{cof}(\alpha)\), \(g: \mathrm{cof}(\alpha) \to \alpha\) with unbounded range, then \(g \circ f: \delta \to \alpha\) also has unbounded range, contradicting minimality of \(\mathrm{cof}(\alpha)\). An ordinal is regular if it is equal to its own cofinality, else it is singular. So: * [[0]], [[1]] and [[Omega|\(\omega\)]] are regular. * All natural numbers other than \(0\) and \(1\) are singular. * All countable infinite ordinals other than \(\omega\) are singular. * \(\mathrm{cof}(\alpha)\) is regular for any \(\alpha\). Cofinality is used in the definition of [[Inaccessible cardinal|weakly inaccessible]] cardinals. ==Without choice== Citation about every uncountable cardinal being singular being consistent with ZF 4ee7cc7116d2b4050eba0fd71d6f760d27cea9a3 0 226 731 691 2024-03-25T16:53:02Z CreeperBomb 30 Undo revision [[Special:Diff/691|691]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Nothing OCF is a weak [[Ordinal collapsing function|OCF]], defined by CatIsFluffy. It is similar to many other OCFs in definition, but omits addition. Therefore, the growth rate is much, much slower. It is believed to correspond to a weak version of [[Extended Buchholz's function]], also defined by omitting addition, and that it catches up to the ordinary version of EBOCF by [[Extended Buchholz ordinal|EBO]]. However, no proof of either of these claims has been given and they remain open questions. fefcbce2a067ab3c9135b9b2763a46590cbe5c59 0 163 732 692 2024-03-25T16:53:24Z CreeperBomb 30 Undo revision [[Special:Diff/692|692]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it with the Hebrew later for Tav and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for [[Reflection principle|reflection principles]], with totally reflecting cardinals probably being the closest first-order object similar to his description, and he explored it more from a philosophical standpoint. In particular, Cantor associated it mathematically with the class of cardinals (not a set by a problem similar to the [[Burali–Forti paradox]]), so large it almost "transcended" itself, and associated it metaphysically with God. Absolute infinity and attempts to define numbers beyond (which is ironic, since the whole point of absolute infinity is that it could not be transcended beyond) feature prominently in fictional googology. ==As justification for reflection== Later authors have connected Cantor's remark that absolute infinity "can not be conceived" to reflection principles. For example, Maddy states:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic, vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces ''reflection'' to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \( V \) is already true of some [\( V_\alpha \)]. 8403130d80e4112d314530c71c1532cf9678d020 0 114 733 693 2024-03-25T16:53:40Z CreeperBomb 30 Undo revision [[Special:Diff/693|693]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma\). The least additive principal ordinal is 1 since \(0 + 0 < 1\), and all additive principal ordinals other than 1 are limit ordinals. In particular, as can be seen from the Cantor normal form theorem (every ordinal has a [[Cantor normal form|CNF]] representation), additive principal ordinals are precisely the ordinals of the form \(\omega^\gamma\) for some ordinal \(\gamma\). As such, the second infinite additive principal ordinal is [[Omega^2|\(\omega^2\)]], the first limit of additive principal ordinals is [[Omega^omega|\(\omega^\omega\)]], and the first limit of the limits of additive principal ordinals is \(\omega^{\omega^2}\). Additive principal ordinals can be generalized to multiplicative principal ordinals and exponential principal ordinals. The former are precisely the ordinals of the form \(\omega^{\omega^\gamma}\) for some \(\gamma\), and one can consider that multiplicative principal ordinals are to additive principal ordinals as additive principal ordinals are to limit ordinals. However, exponential principal ordinals are not ordinals of the form \(\omega^{\omega^{\omega^\gamma}}\) for some \(\gamma\) but, rather, are just \(\omega\) and the [[epsilon numbers]]. 31ac249acf6e959f0a645a566cef14a239a07e07 0 192 734 694 2024-03-25T16:54:15Z CreeperBomb 30 Undo revision [[Special:Diff/694|694]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. [[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The question of whether \(\omega_1\), the least uncountable [[cardinal]], and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the [[continuum hypothesis]]. Surprisingly, assuming its consistency, this is neither provable nor disprovable in [[ZFC]]! The existence of an arbitrary set's powerset is not provable from [[Kripke-Platek set theory|KP]], even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC. eb859b56888463de272b40b3d1c25b466785009f 0 198 735 695 2024-03-25T16:54:38Z CreeperBomb 30 Undo revision [[Special:Diff/695|695]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Hilbert's Grand Hotel is a famous analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the often unintuitive ways infinite [[bijections]] work, it is actually possible to still fit many more people. Firstly, if there is a single new guest who wants a room, it is possible to accommodate by simply telling everyone to move up one room - so the person checked in Room 0 moves to Room 1, the person checked in Room 1 moves to Room 2, and so on. Because every room has a room coming after it, everybody who was checked in still has a room. Yet Room 0 is now empty - the new guest can check in there. This is analogous to the proof that [[Omega|\(\omega\)]] and \(\omega+1\) are equinumerous (that is, they have the same [[cardinality]]). Similarly, if someone in room \(n\) checks out of the hotel, then everybody in room \(m\) for \(m > n\) can move one room to the left, and all the rooms will be filled again. One can also accommodate countably infinitely many new guests, by requiring that every current guest in Room \(n\) goes to Room \(2n\) and that the \(n\)th new guest go to Room \(2n+1\). The first part frees up all the odd-numbered rooms, which the new guests can fill up. Therefore, \(\omega 2\) is equinumerous with \(\omega\). In fact, it's even possible to accommodate a countably infinite collection of countably infinitely many sets of new guests! One can assign the current guest in room \(n\) to room \(2^n\), the \(n\)th guest in the first collection of new guests to room \(3^n\), the \(n\)th guest in the next collection of new guests to room \(5^n\), then \(7^n\), \(11^n\), and so on. Because there are infinitely many prime numbers, and powers of primes never overlap, everybody can be accommodated - even with many rooms now empty, such as room 6, which isn't a power of any prime number! However, not every infinite batch of guests can fit in Hilbert's Grand Hotel. If a bus brings infinitely many guests whose names are all infinite strings made up of "a" and "b", and every string has a guest, not all of the guests can fit. In fact, it's possible to pair up each name to a real number, showing that there are more real numbers than natural numbers, even though there are infinitely many of both! c37ad2299aac8aa11e79b13cb2cd129f83ed8d8a 0 54 736 696 2024-03-25T16:54:52Z CreeperBomb 30 Undo revision [[Special:Diff/696|696]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu: \textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984. ==History== <nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref> == Definition == We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_\mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras. == Extension == This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: * \(C_\nu^0(\alpha) = \Omega_\nu\) * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_\mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) * \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) * \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function. This admits an ordinal notation too, as well as a canonical set of fundamental sequences. == References == 877c52fbc37086276c727c21232a7d9c784f2f99 0 49 737 697 2024-03-25T16:55:03Z CreeperBomb 30 Undo revision [[Special:Diff/697|697]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) wikitext text/x-wiki The '''Burali–Forti paradox''' refers to the theorem that there is no set containing all [[von Neumann ordinal]]s. Essentially, if there were such a set, then it would itself be a von Neumann ordinal, contradicting the axiom of foundation, which implies no set can be an element of itself. In second-order theories such as Morse-Kelley set theory, this issue is circumvented by making the collection of ordinals a proper class, while all ordinals are sets (and proper classes can not contain other proper classes). 539c7d8bf0fe09ba0aaf02d79f5f80fe515e0f8c 10 230 739 2024-03-25T16:56:34Z CreeperBomb 30 Created page with "<div name="Deletion notice" class="boilerplate metadata" id="delete" style="background-color: #fee; margin: 0 1em; padding: 0 10px; border: 1px solid #aaa;"> '''This page is a candidate for deletion. {{#if:{{{1|}}}|<nowiki/> The reason given is: {{{1}}}.|No reason was given for its deletion.}}''' If you disagree with its deletion, please explain why at [[Category talk:Candidates for deletion]] or improve the page and remove the <code>{{t|delete}}</code> tag.{{#ifexist:{..." wikitext text/x-wiki <div name="Deletion notice" class="boilerplate metadata" id="delete" style="background-color: #fee; margin: 0 1em; padding: 0 10px; border: 1px solid #aaa;"> '''This page is a candidate for deletion. {{#if:{{{1|}}}|<nowiki/> The reason given is: {{{1}}}.|No reason was given for its deletion.}}''' If you disagree with its deletion, please explain why at [[Category talk:Candidates for deletion]] or improve the page and remove the <code>{{t|delete}}</code> tag.{{#ifexist:{{TALKPAGENAME}}|<br /><br />There may be a discussion on this page's deletion at [[{{TALKPAGENAME}}|the talk page]].|}} Remember to check [[Special:Whatlinkshere/{{NAMESPACE}}:{{PAGENAME}}|what links here]] and [{{SERVER}}{{localurl:{{NAMESPACE}}:{{PAGENAME}}|action=history}} the page history] before deleting. </div><br/><includeonly>[[Category:Candidates for deletion]]</includeonly><noinclude> To use this template, type <code>{{t|delete}}</code> on the page to be deleted. To provide a reason for the deletion nomination, type <code>{{t|delete|reason}}</code>. {{Deletion templates}} [[Category:Article management templates|{{PAGENAME}}]] </noinclude> 92574f5841f9122fbbca7254121bac9b340122cb 4 231 740 2024-03-25T17:08:19Z CreeperBomb 30 Created page with "== Rules == # All content must be legally contributed, including but not limited to the conditions that: #* All contributions must be public domain and to be released under the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) #* Content may not be explicit or heavily suggestive # Vandalism is not allowed, and is defined as intentional destruction of a page, file, etc. without community agreement for the change, except in the case of removing illeg..." wikitext text/x-wiki == Rules == # All content must be legally contributed, including but not limited to the conditions that: #* All contributions must be public domain and to be released under the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) #* Content may not be explicit or heavily suggestive # Vandalism is not allowed, and is defined as intentional destruction of a page, file, etc. without community agreement for the change, except in the case of removing illegally contributed content # Harassment, doxing, or other attack on a person is not tolerated # By contributing, you agree that your contributions are truthful aa7fd84a61f353bef386f31f8ea4f5990502932d 741 740 2024-03-25T17:09:43Z CreeperBomb 30 wikitext text/x-wiki All users of the Apeirology Wiki agree to these terms and rules in addition to the [https://meta.miraheze.org/wiki/Terms_of_Use Miraheze Terms of Use].   == Rules == # All content must be legally contributed, including but not limited to the conditions that: #* All contributions must be public domain and to be released under the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) #* Content may not be explicit or heavily suggestive # Vandalism is not allowed, and is defined as intentional destruction of a page, file, etc. without community agreement for the change, except in the case of removing illegally contributed content # Harassment, doxing, or other attack on a person is not tolerated # By contributing, you agree that your contributions are truthful 55b90d3a3c6ddb778dfb3a3f47fcda82d5d59777 1 232 742 2024-03-25T17:14:21Z CreeperBomb 30 Created page with "== User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. ~~~~" wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) 3fcafa604df4ef22103571c377739d6bb47ce8e5 743 742 2024-03-25T20:22:33Z Cobsonwabag 32 /* User:Cobsonwabag */ Reply wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) :id like to see proof of that [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 20:22, 25 March 2024 (UTC) af90dbfdddb6c3264d4594ce2a1e8158c5069e66 747 743 2024-03-26T05:08:26Z CreeperBomb 30 /* User:Cobsonwabag */ Reply wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) :id like to see proof of that [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 20:22, 25 March 2024 (UTC) ::https://apeirology.com/wiki/Special:Contributions/Cobsonwabag [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 05:08, 26 March 2024 (UTC) 56e3a57e923069baff94c9805f9e94dc6cd6731d 752 747 2024-03-26T05:15:14Z CreeperBomb 30 wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) :id like to see proof of that [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 20:22, 25 March 2024 (UTC) :: https://apeirology.com/wiki/Special:Contributions/Cobsonwabag. Examples include: https://apeirology.com/wiki/Main_Page?oldid=746, https://apeirology.com/wiki/Proving_well-orderedness?oldid=744, https://apeirology.com/wiki/Burali%E2%80%93Forti_paradox?oldid=697, https://apeirology.com/wiki/Buchholz%27s_psi-functions?oldid=696, and almost every other edit for vandalism, and https://apeirology.com/wiki/File:Coinslot.png?oldid=658 for explicit content [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 05:08, 26 March 2024 (UTC) 6a1d34365ab9130280cdf6b17bbca6fd50794ed8 753 752 2024-03-26T12:20:47Z Cobsonwabag 32 /* User:Cobsonwabag */ Reply wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) :id like to see proof of that [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 20:22, 25 March 2024 (UTC) :: https://apeirology.com/wiki/Special:Contributions/Cobsonwabag. Examples include: https://apeirology.com/wiki/Main_Page?oldid=746, https://apeirology.com/wiki/Proving_well-orderedness?oldid=744, https://apeirology.com/wiki/Burali%E2%80%93Forti_paradox?oldid=697, https://apeirology.com/wiki/Buchholz%27s_psi-functions?oldid=696, and almost every other edit for vandalism, and https://apeirology.com/wiki/File:Coinslot.png?oldid=658 for explicit content [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 05:08, 26 March 2024 (UTC) :::ip grabbers [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 12:20, 26 March 2024 (UTC) 568e548d62e7c7b2b465ffe91995316b172a5c31 0 64 744 706 2024-03-25T20:26:45Z Cobsonwabag 32 Cobsonwabag moved page [[Proving well-orderedness]] to [[User talk:Proving well-orderedness]] wikitext text/x-wiki Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitrarily obscure cases, of course, the proof can be arbitrarily unusual itself. However, for common special cases, there are some shared features of the proof. <h2>[[Expansion system | Expansion systems]]</h2> In an expansion system, totality translates to "for every pair of terms \( x,y \), one is reachable from the other by expansion". The order can usually easily be proven to respect a lexicographical order or some other order known to be total (i.e. one can prove \( x\prec y \) implies \( x<y \), where \( \prec \) is the order of the expansion system and \( < \) is a total order), and then it is simply about proving that, intuitively, repeated expansion is never forced to skip any specific term. <br>Then a common property that directly leads to totality is the conjunction of \( x[n]\preceq x[n+1] \), \( x[n]\prec y\preceq x[n+1]\Rightarrow x[n]\preceq y[0] \), and the statement that iterating \( [0] \) always reaches the minimum eventually, no matter what you start with. The intuitive reason why this implies totality is that if we have \( x<y\prec z \) and \( x=z[n_0][n_1]...[n_m] \), then \( x \) can be reached from \( y \) by repeating \( [0] \) until it reaches something of the form \( z[n_0][n_1]...[n_k][a] \) with \( a>n_{k+1} \), at which point \( [b] \) is used with \( b \) minimal so that this doesn't go below \( x \), and the whole process repeats. The fact that this eventually terminates follows from looking at the \( (n_0,n_1,...,n_k,a) \) that appear that way, and noticing that this decreases lexicographically as the process moves on the sequence always decreases lexicographically and its length is bounded by \( m+1 \), so this form can only be reached finitely many times, and between all that, we're only iterating \( [0] \), which is guaranteed to decrease the term as much as we need, eventually getting us to \( x \) only by expanding \( y \). Keep in mind that this is not a formal proof. <br>It is not always true that this property holds. This page is currently unfinished. 55136a51229ca84711bb531fead496230b69f5b7 748 744 2024-03-26T05:09:28Z CreeperBomb 30 CreeperBomb moved page [[User talk:Proving well-orderedness]] to [[Proving well-orderedness]] over redirect: Stop wikitext text/x-wiki Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitrarily obscure cases, of course, the proof can be arbitrarily unusual itself. However, for common special cases, there are some shared features of the proof. <h2>[[Expansion system | Expansion systems]]</h2> In an expansion system, totality translates to "for every pair of terms \( x,y \), one is reachable from the other by expansion". The order can usually easily be proven to respect a lexicographical order or some other order known to be total (i.e. one can prove \( x\prec y \) implies \( x<y \), where \( \prec \) is the order of the expansion system and \( < \) is a total order), and then it is simply about proving that, intuitively, repeated expansion is never forced to skip any specific term. <br>Then a common property that directly leads to totality is the conjunction of \( x[n]\preceq x[n+1] \), \( x[n]\prec y\preceq x[n+1]\Rightarrow x[n]\preceq y[0] \), and the statement that iterating \( [0] \) always reaches the minimum eventually, no matter what you start with. The intuitive reason why this implies totality is that if we have \( x<y\prec z \) and \( x=z[n_0][n_1]...[n_m] \), then \( x \) can be reached from \( y \) by repeating \( [0] \) until it reaches something of the form \( z[n_0][n_1]...[n_k][a] \) with \( a>n_{k+1} \), at which point \( [b] \) is used with \( b \) minimal so that this doesn't go below \( x \), and the whole process repeats. The fact that this eventually terminates follows from looking at the \( (n_0,n_1,...,n_k,a) \) that appear that way, and noticing that this decreases lexicographically as the process moves on the sequence always decreases lexicographically and its length is bounded by \( m+1 \), so this form can only be reached finitely many times, and between all that, we're only iterating \( [0] \), which is guaranteed to decrease the term as much as we need, eventually getting us to \( x \) only by expanding \( y \). Keep in mind that this is not a formal proof. <br>It is not always true that this property holds. This page is currently unfinished. 55136a51229ca84711bb531fead496230b69f5b7 0 1 746 699 2024-03-25T20:28:20Z Cobsonwabag 32 wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] * [https://soyjak.party List of cardinals (another wiki)] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> fabacb4d04db248406b0077e34f201d2521aa8a0 751 746 2024-03-26T05:11:20Z CreeperBomb 30 Undo revision [[Special:Diff/746|746]] by [[Special:Contributions/Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) Vandalism wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> e2b43ff20f800c007ae08aa90b03c1ad7f9c7348 3 234 749 2024-03-26T05:09:28Z CreeperBomb 30 CreeperBomb moved page [[User talk:Proving well-orderedness]] to [[Proving well-orderedness]] over redirect: Stop wikitext text/x-wiki #REDIRECT [[Proving well-orderedness]] 593db7bd3f8f7f47ca1363a32cf10d778e6ae19e 750 749 2024-03-26T05:09:58Z CreeperBomb 30 Removed redirect to [[Proving well-orderedness]] wikitext text/x-wiki {{delete}} 1b0ce0563f5afcd669fe10969b3140a584202866 0 235 754 2024-03-30T21:16:22Z CreeperBomb 30 Copied from wikipedia https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_arithmetic wikitext text/x-wiki We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic. == Successor cardinal == If the axiom of choice holds, then every cardinal κ has a successor, denoted κ+, where κ+ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that \(κ^+≰κ\). For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal. == Cardinal addition == If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}). : \( |X|+|Y|=|X\cup Y|.\) * Zero is an additive identity κ + 0 = 0 + κ = κ. * Addition is associative (κ + μ) + ν = κ + (μ + ν). * Addition is commutative κ + μ = μ + κ. * Addition is non-decreasing in both arguments: \( (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu ))\). Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either κ or μ is infinite, then : \( \kappa +\mu =\max\{\kappa ,\mu \}\,\). === Subtaction === Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ. == Cardinal multiplication == The product of cardinals comes from the Cartesian product: : \( |X|\cdot |Y|=|X\times Y|\) * κ·0 = 0·κ = 0. * κ·μ = 0 → (κ = 0 or μ = 0). * One is a multiplicative identity: κ·1 = 1·κ = κ. * Multiplication is associative (κ·μ)·ν = κ·(μ·ν). * Multiplication is commutative κ·μ = μ·κ. * Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ). * Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ. Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then : \( \kappa \cdot \mu =\max\{\kappa ,\mu \}\). === Division === Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π. == Cardinal exponentiation == Exponentiation is given by :\( |X|^{|Y|}=\left|X^{Y}\right|\), where XY is the set of all functions from Y to X. * κ⁰ = 1 (in particular 0⁰ = 1), see empty function. * If 1 ≤ μ, then 0^μ = 0. * 1^μ = 1. * κ¹ = κ. * κ^{μ + ν} = κ^μ·κ^ν. * κ^{μ · ν} = (κ^μ)^ν. * (κ·μ)^ν = κ^ν·μ^ν. Exponentiation is non-decreasing in both arguments: * (1 ≤ ν and κ ≤ μ) → (ν^κ ≤ ν^μ) * (κ ≤ μ) → (κ^ν ≤ μ^ν). 2^|X| is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2^κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.) All the remaining propositions in this section assume the axiom of choice: * If κ and μ are both finite and greater than 1, and ν is infinite, then κ^ν = μ^ν. * If κ is infinite and μ is finite and non-zero, then κ^μ = κ. If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then: * Max (κ, 2^μ) ≤ κ^μ ≤ Max (2^κ, 2^μ). Using König's theorem, one can prove κ < κ^cf(κ) and κ < cf(2^κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ. === Roots === Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying \( \nu ^{\mu }=\kappa \) will be \(\kappa \). === Logarithms === Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \( \mu ^{\lambda }=\kappa \). However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy \( \nu ^{\lambda }=\kappa \). The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2^μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess. 4b9af340291ee9c1a6b6bf72d65bf07dae747a04 755 754 2024-03-30T21:17:22Z CreeperBomb 30 /* Successor cardinal */ wikitext text/x-wiki We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic. == Successor cardinal == If the axiom of choice holds, then every cardinal κ has a successor, denoted κ⁺, where κ⁺ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ⁺ such that \(κ^+≰κ\). For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal. == Cardinal addition == If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}). : \( |X|+|Y|=|X\cup Y|.\) * Zero is an additive identity κ + 0 = 0 + κ = κ. * Addition is associative (κ + μ) + ν = κ + (μ + ν). * Addition is commutative κ + μ = μ + κ. * Addition is non-decreasing in both arguments: \( (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu ))\). Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either κ or μ is infinite, then : \( \kappa +\mu =\max\{\kappa ,\mu \}\,\). === Subtaction === Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ. == Cardinal multiplication == The product of cardinals comes from the Cartesian product: : \( |X|\cdot |Y|=|X\times Y|\) * κ·0 = 0·κ = 0. * κ·μ = 0 → (κ = 0 or μ = 0). * One is a multiplicative identity: κ·1 = 1·κ = κ. * Multiplication is associative (κ·μ)·ν = κ·(μ·ν). * Multiplication is commutative κ·μ = μ·κ. * Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ). * Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ. Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then : \( \kappa \cdot \mu =\max\{\kappa ,\mu \}\). === Division === Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π. == Cardinal exponentiation == Exponentiation is given by :\( |X|^{|Y|}=\left|X^{Y}\right|\), where XY is the set of all functions from Y to X. * κ⁰ = 1 (in particular 0⁰ = 1), see empty function. * If 1 ≤ μ, then 0^μ = 0. * 1^μ = 1. * κ¹ = κ. * κ^{μ + ν} = κ^μ·κ^ν. * κ^{μ · ν} = (κ^μ)^ν. * (κ·μ)^ν = κ^ν·μ^ν. Exponentiation is non-decreasing in both arguments: * (1 ≤ ν and κ ≤ μ) → (ν^κ ≤ ν^μ) * (κ ≤ μ) → (κ^ν ≤ μ^ν). 2^|X| is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2^κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.) All the remaining propositions in this section assume the axiom of choice: * If κ and μ are both finite and greater than 1, and ν is infinite, then κ^ν = μ^ν. * If κ is infinite and μ is finite and non-zero, then κ^μ = κ. If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then: * Max (κ, 2^μ) ≤ κ^μ ≤ Max (2^κ, 2^μ). Using König's theorem, one can prove κ < κ^cf(κ) and κ < cf(2^κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ. === Roots === Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying \( \nu ^{\mu }=\kappa \) will be \(\kappa \). === Logarithms === Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \( \mu ^{\lambda }=\kappa \). However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy \( \nu ^{\lambda }=\kappa \). The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2^μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess. 58fdb143a720915d6c1d4a51087949b178af7c7f 756 755 2024-03-30T21:17:41Z CreeperBomb 30 /* Successor cardinal */ wikitext text/x-wiki We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic. == Successor cardinal == If the axiom of choice holds, then every cardinal κ has a successor, denoted κ⁺, where κ⁺ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ⁺ such that \(κ^+≰κ\)). For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal. == Cardinal addition == If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}). : \( |X|+|Y|=|X\cup Y|.\) * Zero is an additive identity κ + 0 = 0 + κ = κ. * Addition is associative (κ + μ) + ν = κ + (μ + ν). * Addition is commutative κ + μ = μ + κ. * Addition is non-decreasing in both arguments: \( (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu ))\). Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either κ or μ is infinite, then : \( \kappa +\mu =\max\{\kappa ,\mu \}\,\). === Subtaction === Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ. == Cardinal multiplication == The product of cardinals comes from the Cartesian product: : \( |X|\cdot |Y|=|X\times Y|\) * κ·0 = 0·κ = 0. * κ·μ = 0 → (κ = 0 or μ = 0). * One is a multiplicative identity: κ·1 = 1·κ = κ. * Multiplication is associative (κ·μ)·ν = κ·(μ·ν). * Multiplication is commutative κ·μ = μ·κ. * Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ). * Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ. Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then : \( \kappa \cdot \mu =\max\{\kappa ,\mu \}\). === Division === Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π. == Cardinal exponentiation == Exponentiation is given by :\( |X|^{|Y|}=\left|X^{Y}\right|\), where XY is the set of all functions from Y to X. * κ⁰ = 1 (in particular 0⁰ = 1), see empty function. * If 1 ≤ μ, then 0^μ = 0. * 1^μ = 1. * κ¹ = κ. * κ^{μ + ν} = κ^μ·κ^ν. * κ^{μ · ν} = (κ^μ)^ν. * (κ·μ)^ν = κ^ν·μ^ν. Exponentiation is non-decreasing in both arguments: * (1 ≤ ν and κ ≤ μ) → (ν^κ ≤ ν^μ) * (κ ≤ μ) → (κ^ν ≤ μ^ν). 2^|X| is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2^κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.) All the remaining propositions in this section assume the axiom of choice: * If κ and μ are both finite and greater than 1, and ν is infinite, then κ^ν = μ^ν. * If κ is infinite and μ is finite and non-zero, then κ^μ = κ. If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then: * Max (κ, 2^μ) ≤ κ^μ ≤ Max (2^κ, 2^μ). Using König's theorem, one can prove κ < κ^cf(κ) and κ < cf(2^κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ. === Roots === Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying \( \nu ^{\mu }=\kappa \) will be \(\kappa \). === Logarithms === Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \( \mu ^{\lambda }=\kappa \). However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy \( \nu ^{\lambda }=\kappa \). The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2^μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess. cfa9a98bf224e4316d185723c31cb7d90d4f1456 3 236 757 2024-03-30T21:18:28Z CreeperBomb 30 Created page with "[https://googology.miraheze.org/wiki/User:CreeperBomb Go here for user page] [https://googology.miraheze.org/wiki/User_talk:CreeperBomb Go here for talk page]" wikitext text/x-wiki [https://googology.miraheze.org/wiki/User:CreeperBomb Go here for user page] [https://googology.miraheze.org/wiki/User_talk:CreeperBomb Go here for talk page] 5d024c26b25d8d00251ac425cdf00a222bd7cc9d 10 237 758 2024-05-21T14:55:30Z CreeperBomb 30 Created page with "^{<nowiki>[</nowiki>[www.apeirology.com/wiki/{{FULLPAGENAME}}?action=edit/ ''Citation needed'']<nowiki>]</nowiki>}" wikitext text/x-wiki ^{<nowiki>[</nowiki>[www.apeirology.com/wiki/{{FULLPAGENAME}}?action=edit/ ''Citation needed'']<nowiki>]</nowiki>} 23415ca8724934da2495cfbd31e2f68f2fcc76a5 759 758 2024-05-21T14:57:16Z CreeperBomb 30 Fixed? wikitext text/x-wiki ^{<nowiki>[</nowiki>[www.apeirology.com/wiki/{{urlencode:{{FULLPAGENAME}}}?action=edit/ ''Citation needed'']<nowiki>]</nowiki>} cddec71c2cb9ae6af7a27e2af37414f8fd3273d5 760 759 2024-05-21T15:01:29Z CreeperBomb 30 Fixed. wikitext text/x-wiki ^{[[{{canonicalurl:{{FULLPAGENAME}}|action=edit}} ''Citation needed'']]} 9ae7d290b6a0c7f0d056ceac6d66c86ae9a8e556 10 238 761 2024-05-21T15:03:34Z CreeperBomb 30 Created page with "{{Template:Template link}}" wikitext text/x-wiki {{Template:Template link}} db49a9699777b808cbba6d43c3fee20b484d6ce6 763 761 2024-05-21T15:04:31Z CreeperBomb 30 Fixed wikitext text/x-wiki {{template link}} 71d135f5fae9887cdfeb15b6fc05d87cdb4c6875 10 239 762 2024-05-21T15:03:57Z CreeperBomb 30 Created page with "A template link with a variable number of parameters (0-20). Use: :{{template link|template link|parameter1|parameter2|parameter3|parameter4|...|parameter20}} <!-- self-referential examples! --> Aliases: * {{Template link|t}} * {{Template link|tl}} * {{Template link|tp}} * {{Template link|template}} Example: <code><nowiki>{{t|Welcome}}</nowiki></code> :{{t|Welcome}} <code><nowiki>{{t|Welcome|Item1|Item2|Item3|Item4|Item5|...}}</nowiki></code> :{{t|Welcome|Item1|Item..." wikitext text/x-wiki A template link with a variable number of parameters (0-20). Use: :{{template link|template link|parameter1|parameter2|parameter3|parameter4|...|parameter20}} <!-- self-referential examples! --> Aliases: * {{Template link|t}} * {{Template link|tl}} * {{Template link|tp}} * {{Template link|template}} Example: <code><nowiki>{{t|Welcome}}</nowiki></code> :{{t|Welcome}} <code><nowiki>{{t|Welcome|Item1|Item2|Item3|Item4|Item5|...}}</nowiki></code> :{{t|Welcome|Item1|Item2|Item3|Item4|Item5|...}} Template: :<onlyinclude><nowiki>{{</nowiki>[[Template:{{{1}}}|{{{1}}}]]{{t/piece|{{{2|---}}}}}{{t/piece|{{{3|---}}}}}{{t/piece|{{{4|---}}}}}{{t/piece|{{{5|---}}}}}{{t/piece|{{{6|---}}}}}{{t/piece|{{{7|---}}}}}{{t/piece|{{{8|---}}}}}{{t/piece|{{{9|---}}}}}{{t/piece|{{{10|---}}}}}{{t/piece|{{{11|---}}}}}{{t/piece|{{{12|---}}}}}{{t/piece|{{{13|---}}}}}{{t/piece|{{{14|---}}}}}{{t/piece|{{{15|---}}}}}{{t/piece|{{{16|---}}}}}{{t/piece|{{{17|---}}}}}{{t/piece|{{{18|---}}}}}{{t/piece|{{{19|---}}}}}{{t/piece|{{{20|---}}}}}{{t/piece|{{{21|---}}}}}<nowiki>}}</nowiki></onlyinclude> See also: [[wikipedia:Template:Template Link]] [[Category:General wiki templates|Template link]] 4fda4e5e8dcfe4c0d676038dc892d431a78f24b5 10 240 764 2024-05-21T15:04:55Z CreeperBomb 30 Created page with "Parameter piece for {{t|t}}, controls styling/showing of parameter fragments. Template (invisible by design): :<onlyinclude>{{#ifeq: {{{1|---}}}|---||&#124;<font color="gray">''&lt;{{{1}}}&gt;''</font>}}</onlyinclude> [[Category:General wiki templates|Template link parameter piece]]" wikitext text/x-wiki Parameter piece for {{t|t}}, controls styling/showing of parameter fragments. Template (invisible by design): :<onlyinclude>{{#ifeq: {{{1|---}}}|---||&#124;<font color="gray">''&lt;{{{1}}}&gt;''</font>}}</onlyinclude> [[Category:General wiki templates|Template link parameter piece]] 03c7300fde39a450225af325e8e08ad2f8cf6d95 0 241 765 2024-05-21T15:21:47Z CreeperBomb 30 Created page with "{{stub}} A [[matchstick diagram]] is a particular representation for ordinals. The idea is to lay down vertical lines left-to-right such that the diagram has an order type of the ordinal, where being left of a stick is equivalent to being lesser. Equivalently, enumerating a diagram will result in all ordinals less than the corresponding ordinal having their own stick. This second definition is simply another way of saying the first definition. Matchsticks may have unequa..." wikitext text/x-wiki {{stub}} A [[matchstick diagram]] is a particular representation for ordinals. The idea is to lay down vertical lines left-to-right such that the diagram has an order type of the ordinal, where being left of a stick is equivalent to being lesser. Equivalently, enumerating a diagram will result in all ordinals less than the corresponding ordinal having their own stick. This second definition is simply another way of saying the first definition. Matchsticks may have unequal heights in order to make the diagram easier to read. Although matchstick diagrams can have infinitely many sticks, they are usually restricted to a finite region to make reading easier, as infinite area offers no special advantage. Matchstick diagrams have the property that placing two diagrams side-by-side results in the sum of their corresponding ordinals. For example, placing the diagram for 1 before \(\omega\) results in the diagram \(1 + \omega = \omega\), and placing \(\omega\) before 1 results in \(\omega + 1 = \omega + 1 \). Another property of matchstick diagrams is that they can only represent countable ordinals. de0dd77ffecf74e769c9931653145cab154a03f6 0 156 766 585 2024-05-31T21:10:15Z C7X 9 Small cardinals under AD wikitext text/x-wiki The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the [[axiom of choice]], since it implies that there is no [[Well-ordered set|well-ordering]] of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is very high. Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). Assuming AD, \(\aleph_1\) and \(\aleph_2\) are [[Measurable cardinal|measurable]], \(\aleph_n\) is singular for all \(2<n<\omega\), and \(\aleph_{\omega+1}\) is measurable.<ref>T. Jech, "About the Axiom of Choice". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref>^{p. 369} 8fca8e3dbc15f0fbcf23536d3cf51dd3ff603fa6 767 766 2024-05-31T21:10:29Z C7X 9 wikitext text/x-wiki The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the [[axiom of choice]], since it implies that there is no [[Well-ordered set|well-ordering]] of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is very high. Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). Assuming AD, \(\aleph_1\) and \(\aleph_2\) are [[Measurable cardinal|measurable]], \(\aleph_n\) is singular for all \(2<n<\omega\), and \(\aleph_{\omega+1}\) is measurable.<ref>T. Jech, "About the Axiom of Choice". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref>^{p. 369} ==References== <reflist /> 74c5f5b844c9f10d8df8b7ddf5a27d4e2b98d2c2 768 767 2024-05-31T21:10:44Z C7X 9 /* References */ wikitext text/x-wiki The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states that, for every subset \(A\) of Baire space, one of the two players has a winning strategy in \(\mathcal{G}_A\). AD is known to be inconsistent with the [[axiom of choice]], since it implies that there is no [[Well-ordered set|well-ordering]] of the real numbers. However, its consistency strength relative to [[ZFC|\(\mathrm{ZF}\)]] is very high. Note that the determinacy of every topological game whose payoff set is closed, or even Borel, is already provable in \(\mathrm{ZFC}\). Sufficient large cardinal axioms imply that every game with projective, or even quasi-projective, payoff set is determined, while still remaining consistent with the axiom of choice. By a theorem of Woodin, \(\mathrm{ZF} + \mathrm{AD}\) is equiconsistent with \(\mathrm{ZFC} + \mathrm{PD}\), where \(\mathrm{PD}\) is the assertion that every topological game with projective payoff set is determined, which is equiconsistent \(\mathrm{ZFC}\) augmented by the existence of infinitely many Woodin cardinals. Since Woodin cardinals are [[Mahlo cardinal|strongly Mahlo]], if the axiom of determinacy is consistent, then so is the existence of infinitely many Mahlo cardinals. Furthermore, let \(L(\mathbb{R})\) be the smallest [[Inner model theory|inner model]] containing both all [[Ordinal|ordinals]] and all real numbers. Then the existence of both infinitely many Woodin cardinals and a [[measurable]] cardinal above them implies that \(L(\mathbb{R})\) does not satisfy the axiom of choice but, rather the axiom of determinacy. Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\). Assuming AD, \(\aleph_1\) and \(\aleph_2\) are [[Measurable cardinal|measurable]], \(\aleph_n\) is singular for all \(2<n<\omega\), and \(\aleph_{\omega+1}\) is measurable.<ref>T. Jech, "About the Axiom of Choice". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref>^{p. 369} 8fca8e3dbc15f0fbcf23536d3cf51dd3ff603fa6 0 121 769 643 2024-06-06T20:49:55Z C7X 9 /* Justification and motivation for large cardinal axioms */ wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, J. Reitz, R. Schindler, [https://arxiv.org/abs/1708.06669 Inner-model reflection principles] (2018). Accessed 4 September 2023.</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-[[Correct cardinal|correct]]. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give some informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. ==References== 699ff676f204c44cb751a3e0b656abadcb47b822 770 769 2024-06-06T20:50:37Z C7X 9 /* Justification and motivation for large cardinal axioms */ wikitext text/x-wiki A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms. ==Levy-Montague reflection== One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, J. Reitz, R. Schindler, [https://arxiv.org/abs/1708.06669 Inner-model reflection principles] (2018). Accessed 4 September 2023.</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory. Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-[[Correct cardinal|correct]]. An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere) This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example. == Justification and motivation for large cardinal axioms == Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref>^p.503 : Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)]. As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give some informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" /> ===Examples of ordinal properties from reflection principles=== Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\). By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory. Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals. A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following: A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable. ==References== 0f48841ece2e64fa3193be3d5a2c3b50e183e223 2 2 771 512 2024-06-26T18:51:52Z Augigogigi 2 wikitext text/x-wiki Hello! I'm Augigogigi, you may know me as TGR or Augi. [[User:Augigogigi/SbOCF|SbOCF]] 74f59101c737a24ba85e9ec3676f8e058e87bcc0 0 179 772 539 2024-08-10T04:42:43Z C7X 9 Could have been read as "if kappa is measurable, kappa is not in L" wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]], but not \(\Sigma^2_1\)-indescribable. This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be measurable in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are compact.<ref>https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large. 3bf170ac49391b118d0bc740dd91d9a6ca7ccb6f 785 772 2024-11-03T11:58:38Z C7X 9 wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]], but not \(\Sigma^2_1\)-indescribable. This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be measurable in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are compact.<ref>https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large. ==Sources== <references/> e81d1b250cda873f1d371f5ca3565ea666c5230c 786 785 2024-11-03T12:11:13Z C7X 9 wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]],<ref name="Drake74">F. R. Drake, ''Set Theory: An Introduction to Large Cardinals''. Studies in Logic and the Foundations of Mathematics vol. 76 (1974), North-Holland Publishing Company, ISBN 0 7204 2200 0.</ref>^{p. 281} but the least measurable cardinal is not \(\Sigma^2_1\)-indescribable.<ref name="Drake74" />^{p. 283} This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be measurable in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are compact.<ref>https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large. ==Sources== <references/> 61f2e372c238b53d74c0d8dc4320ba93104911aa 787 786 2024-11-03T12:15:09Z C7X 9 wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]],<ref name="Drake74">F. R. Drake, ''Set Theory: An Introduction to Large Cardinals''. Studies in Logic and the Foundations of Mathematics vol. 76 (1974), North-Holland Publishing Company, ISBN 0 7204 2200 0.</ref>^{p. 281} but the least measurable cardinal is not \(\Sigma^2_1\)-indescribable.<ref name="Drake74" />^{p. 283} This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be measurable in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are weakly compact.<ref>K. Millar, "[https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 A weakening of cardinal compactness - is it equivalent?]" (2018). MathOverflow question, accessed 1 September 2023.</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then "almost all" cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is large. ==Sources== <references/> ff8bf420ba031dcb13243e689b29e2dcdd25920a 796 787 2024-11-09T11:41:23Z DV103 40 (1) changed '"almost all"' to 'a lot of' as "most" cardinals below κ are still just successor cardinals (2) put quotation marks around 'large' to make clear that this is meant intuitively and is not a rigorous mathematical concept (maybe this can be changed to 'stationary' instead) wikitext text/x-wiki A measurable cardinal is a certain type of [[large cardinal]] which possesses strong properties. It was one of the first large cardinal axioms to be developed, after [[Inaccessible cardinal|inaccessible]], [[Mahlo cardinal|Mahlo]] and [[Weakly compact cardinal|weakly compact]] cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is [[Indescribable cardinal|\(\Pi^2_1\)-indescribable]],<ref name="Drake74">F. R. Drake, ''Set Theory: An Introduction to Large Cardinals''. Studies in Logic and the Foundations of Mathematics vol. 76 (1974), North-Holland Publishing Company, ISBN 0 7204 2200 0.</ref>^{p. 281} but the least measurable cardinal is not \(\Sigma^2_1\)-indescribable.<ref name="Drake74" />^{p. 283} This doesn't make it weaker than shrewd cardinals, since the correlation between size and consistency strength breaks down at the level of large large cardinals. Measurable cardinals were the subject of Scott's famous proof that if measurable cardinals existed, then [[Constructible hierarchy|\(V \neq L\)]]: therefore, if a cardinal is measurable, it won't be measurable in \(L\), which can be explained by the fact that the objects necessary to show a measurable cardinal is measurable would not be contained within \(L\). There are multiple equivalent definitions of measurability, including one in terms of compactness which shows why measurable cardinals are weakly compact.<ref>K. Millar, "[https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937 A weakening of cardinal compactness - is it equivalent?]" (2018). MathOverflow question, accessed 1 September 2023.</ref> The original definition was that there is a nontrivial ultrafilter on the subsets of \(\kappa\), picking out exactly which subsets of \(\kappa\) are large and which are not, which is closed under \(< \kappa\)-sized intersections. This is a generalization of the existence of a nontrivial ultrafilter on [[Aleph 0|\(\aleph_0\)]], like how inaccessible cardinals are a generalization of \(\aleph_0\) being inaccessible from [[finite]] numbers. If \(\kappa\) is measurable, then a lot of cardinals below \(\kappa\) are strongly inaccessible, strongly Mahlo, weakly compact, Ramsey and more, in the sense that the set of those is "large". ==Sources== <references/> 7a34d0765d3670ca05d1de9d11fabcd36e05f15d 0 15 773 705 2024-08-14T18:42:43Z Racheline 37 fixed some things wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by BashicuHyudora. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of tuples). Then if we let \( m_0 \) be maximal such that the last element \( x \) of the sequence in \( A \) has an \( m_0 \)-parent \( r \), \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from \( r \) to the element right before \( x \), and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The \( i-1 \)-st copy of the \( m \)-parent of \( x \) if \( C = r \) and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), Then, we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> 1b704660176db6baf6f3df0e485f638da4188e10 774 773 2024-08-15T15:18:38Z CreeperBomb 30 Minor spelling mistake wikitext text/x-wiki '''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by BashicuHyudora. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]]. The arrays, however, are only a concise encoding of a deeper underlying structure. In reality, BMS is about structures called "respecting forests" - sequences of elements with infinitely many "ancestry" relations. <h2>Original definition</h2> BMS is an [[expansion system]] with the base of the standard form being \( \{(\underbrace{0,0,...,0,0}_n)(\underbrace{1,1,...,1,1}_n) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way: # The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \). # If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty. # Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \). # \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation. <h2>Interpretation</h2> The definition uses parenthood and ancestry extensively, and can in fact be restated entirely in terms of that. The numbers are only there to encode this structure, similarly to how the numbers in [[Primitive Sequence System]] are only there to encode the hydra. Instead of considering individual entries and their parents/ancestors, it may be easier to consider a whole column \( C \) and its \( m \)-parent/\( m \)-ancestor for each \( m\in\mathbb{N} \), meaning the column containing the parent/ancestor of the \( m \)-th number in \( C \). So this way, we have a structure \( A \) consisting of a finite sequence of elements (each represented by a column), and an infinite sequence of partial orders (\( m \)-ancestry), each partial order respecting the one before, and all of them respecting the order in which the elements appear in the sequence (a relation \( R \) respects a relation \( R' \) if \( R(x_1,x_2,...,x_n)\Rightarrow R'(x_1,x_2,...,x_n) \) for all \( x_1,x_2,...,x_n \), or equivalently, if \( R\subseteq R' \) using the usual encoding of relations as sets of tuples). Then if we let \( m_0 \) be maximal such that the last element \( x \) of the sequence in \( A \) has an \( m_0 \)-parent \( r \), \( A[n] \) is the structure obtained from \( A \) by replacing the last element with \( n \) copies of the elements from \( r \) to the element right before \( x \), and letting the \( m \)-parent of the \( i \)-th copy of an element \( C \) be:<br>- The \( i \)-th copy of the \( m \)-parent of \( C \), if the \( m \)-parent of \( C \) is among the copied elements.<br>- The \( i-1 \)-st copy of the \( m \)-parent of \( x \) if \( C = r \) and \( m<m_0 \).<br>- The \( m \)-parent of \( C \) otherwise. The equivalence of this and the original definition is essentially lemma 2.5 from the claimed proof of well-foundedness.<ref name=":0" /> It can also be restated as a reflection property.^{(to be clarified)} <h2>Well-orderedness and order types</h2> For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref name=":0">[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension. BMS is expected to reach ordinals as high as a good [[ordinal collapsing function]] for ordinals that are \( \alpha-\Sigma_n- \)stable for some \( \alpha\in Ord \) and \( n\in\mathbb{N} \). However, because no such function has been defined yet, this is currently unprovable, considering the informal use of "good". The largest array for which an explicit value was proven is \( (0,0,0)(1,1,1) \), and that value is \( \psi(\Omega_\omega) \) using [[Buchholz's ordinal collapsing function | Buchholz's OCF]].<ref>[https://googology.fandom.com/ja/wiki/ユーザーブログ:P進大好きbot/ペア数列の停止性 Analysis of Pair sequence system]</ref> <h2>Conversion algorithms</h2> Note that the correctness of algorithms further than \((0,0,0)(1,1,1)\) is not proven. Let \(\varepsilon\) denote the empty array, and \(o(A)\) denote the converting-to-ordinals function. <h3>Up to \(\varepsilon_0\)</h3> # \(o(\varepsilon) = 0\). # If we have an array \(A\), then we must have \(A = (0)A_0(0)A_1(0)A_2...(0)A_n\) for positive \(n\), where each of the \(A_i\) do not contain \((0)\) columns. Then, \(o(A) = \omega^{o(A_0^*)}+\omega^{o(A_1^*)}+...+\omega^{o(A_n^*)}\), where \(A^*\) denotes \(A\) with the first entries of each of its columns reduced by one. <h2>References</h2> <references /> 43cf155658c14a2c0b8f43709a840f103c66743f 0 70 775 428 2024-08-18T03:07:55Z CreeperBomb 30 Moved property from out of list into list wikitext text/x-wiki The infinite time Turing machines are a powerful method of computation introduced by Joel David Hamkins and Andy Lewis.<ref>Infinite Time Turing Machines, Joel David Hamkins and Andy Lewis, 1998</ref> They augment the normal notion of a Turing machine (first introduced by Alan Turing in his seminal paper<ref>Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". ''Proceedings of the London Mathematical Society''.</ref>), to a hypothetical machine model which can run for infinitely many steps. It separates the tape into three separate tapes - the input, scratch and output tapes. It's possible to define an analogue of the busy beaver function for ITTMs, denoted \(\Sigma_\infty\), which grows significantly faster than the ordinary busy beaver function, even with an oracle, as well as a halting problem for ITTMs, which has higher Turing degree than \(0^{(\alpha)}\) for all \(\alpha < \gamma\), where \(\gamma\) is the second of the following large ITTM-related ordinals: * \(\lambda\) is the supremum of all ordinals which are the output of an ITTM with empty input. * \(\gamma\) is the supremum of all halting times of an ITTM with empty input. * \(\zeta\) is the supremum of all ordinals which are the eventual output of an ITTM with empty input. * \(\Sigma\) is the supremum of all ordinals which are the accidental output of an ITTM with empty input. ITTMs are quote potent computational models, as they are able to decide whether a given relation on \(\mathbb N\) is a well-order or not.<ref>J. D. Hamkins, A. Lewis, "[https://arxiv.org/abs/math/9808093 Infinite Time Turing Machines]", arXiv 9808093 (2008). Accessed 1 September 2023.</ref>^{theorem 2.2} The ITTM theorem says that: * \(\lambda = \gamma\) * \(\lambda\) is \(\zeta\)-stable (i.e. \(\zeta\)-\(\Sigma_1\)-stable). * \(\zeta\) is the least ordinal which is \(\rho\)-\(\Sigma_2\)-stable for some \(\rho > \zeta\). * \(\Sigma\) is the least ordinal so that \(\zeta\) is \(\Sigma\)-\(\Sigma_2\)-stable. <nowiki>This further shows the computational potency of ITTMs, since the limit of the order-types of well-orders they can compute is much greater than that of normal TMs, i.e. \(\omega_1^{\mathrm{CK}}\).</nowiki> Infinite time Turing machines can themselves be generalized further to \(\Sigma_n\)-machines, with \(\Sigma_2\)-machines being the same as the original.<ref>Friedman, Sy-David & Welch, P. D. (2011). Hypermachines. Journal of Symbolic Logic</ref> d0119d5be44ae6a52cc561d4b6925ab916c28073 6 242 776 2024-08-18T03:23:12Z CreeperBomb 30 A matchstick representation of the ordinal ω2. Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers. wikitext text/x-wiki == Summary == A matchstick representation of the ordinal ω2. Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers. 77e59ac00c1a7774c0fb8be95a9889407881a8e1 777 776 2024-08-18T03:23:42Z CreeperBomb 30 wikitext text/x-wiki == Summary == A matchstick representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers. 55a091ee9441df9be66137dea30ce3d58b6e534c 0 241 778 765 2024-08-18T03:25:55Z CreeperBomb 30 Minor expansion wikitext text/x-wiki {{stub}} A [[matchstick diagram]] is a particular representation for ordinals. The idea is to lay down vertical lines left-to-right such that the diagram has an order type of the ordinal, where being left of a stick is equivalent to being lesser. Equivalently, enumerating a diagram will result in all ordinals less than the corresponding ordinal having their own stick. This second definition is simply another way of saying the first definition. Matchsticks may have unequal heights in order to make the diagram easier to read. Although matchstick diagrams can have infinitely many sticks, they are usually restricted to a finite region to make reading easier, as infinite area offers no special advantage. Matchstick diagrams have the property that placing two diagrams side-by-side results in the sum of their corresponding ordinals. For example, placing the diagram for 1 before \(\omega\) results in the diagram \(1 + \omega = \omega\), and placing \(\omega\) before 1 results in \(\omega + 1 = \omega + 1 \). Additionally, multiplication of \(\alpha\) times \(\beta\) can be computed by replacing all the lines in \(\beta\) by \(\alpha\). Another property of matchstick diagrams is that they can only represent countable ordinals. [[File:Omegasquared.png]] Above is a matchstick diagram for \(\omega^2\) provided as an example. b509b2dc201263b44b3ec85dad6fe3452d8bf853 791 778 2024-11-07T01:31:10Z CreeperBomb 30 Reformatted it wikitext text/x-wiki {{stub}} A [[matchstick diagram]] is a particular representation for ordinals. The idea is to lay down vertical lines left-to-right such that the diagram has the [[order type]] of the ordinal, where being left of a stick is equivalent to being lesser. Equivalently, enumerating a diagram will result in all ordinals less than the corresponding ordinal having their own stick. This second definition is simply another way of saying the first definition. Matchsticks may have unequal heights in order to make the diagram easier to read. Although matchstick diagrams can have infinitely many sticks spread across any sized space, they are usually restricted to a finite region to make reading easier, as infinite area offers no special advantage. Matchstick diagrams have multiple properties: * Placing two diagrams side-by-side results in the sum of their corresponding ordinals. For example, placing the diagram for 1 before \(\omega\) results in the diagram \(1 + \omega = \omega\), and placing \(\omega\) before 1 results in \(\omega + 1 = \omega + 1 \) * Multiplication of \(\alpha\) times \(\beta\) can be computed by replacing all the lines in \(\beta\) by \(\alpha\). Note that due to noncommutativity, this may be different than \(\alpha\) replaced using \(\beta\) * Matchstick diagrams can only represent countable ordinals, as a diagram of an uncountable ordinal would imply a fundamental sequence for \(\omega_1\) [[File:Omegasquared.png]] Above is a matchstick diagram for \(\omega^2\) provided as an example. 44ed43717edd0092d3242d01e78424ec36ad45cf 0 172 779 704 2024-08-21T05:49:19Z C7X 9 Pasting message for reference wikitext text/x-wiki Taranovsky's ordinal notations are a collection of [[Ordinal notation system|ordinal notation systems]] invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and other aspects differ from version to version. These were conjectured originally to be very strong, with the main system possibly reaching the full strength of [[second-order arithmetic]] and beyond. However, it is believed that, due to missing some [[Gandy ordinal|bad ordinal]] structure and other issues, the system may not even reach the subsystem of \(\Pi^1_2\)-comprehension. One of the systems, '''MP''' (Main System with Passthrough), is known to be ill-founded.<ref>Discord message in #taranovsky-notations, reading: <code>00ZZC0Z0ZZCCZCCCC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCCCC > 00ZZC0Z0ZZCCZCCCC0Z0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCCZCCCCCCCCCCCCCCCC > 00ZZC0Z0ZZCCZCCCC0Z0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCCZCCCCCCCCCCCCCCCCCCCCC > 00ZZC0Z0ZZCCZCCCC0Z0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCCZCCCCCCCCCCCCCCCCCCCCCCCCCC > 00ZZC0Z0ZZCCZCCCC0Z0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCC0ZZC0Z0ZZCCZCCCCCCCCZ0ZZCC0ZZC0Z0ZZCCZCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC > ...</code></ref> == DoRI == Degrees of Recursive Inaccessibility are a relatively weak system, compared to the others. Their limit is a recursively hyper-inaccessible cardinal. It uses a system of degrees so that: * The term \(C(a,b,c)\) has admissibility degree \(a\). * Every ordinal has admissibility degree \(0\). * Ordinals of admissibility degree \(1\) are the recursively inaccessible ordinals. * For \(a > 0\), ordinals of admissibility degree \(a+1\) are the ordinals which have admissibility degree \(a\) and are a limit of those. * For limit \(a\), having admissibility degree \(a\) is the same as having every admissibility degree below \(a\). == DoR == Degrees of Reflection are a stronger system. An obsolete analysis suggested their limit was an ordinal \(\alpha\) that is [[Stability|\(\alpha^{++}\)-stable]]; however, a newer analysis suggested that their actual limit is the least bad ordinal, which is significantly smaller. It introduces an ordinal term \(\Omega\) and combines the \(C\)-function with a notation system \(\mathbf{O}\) for ordinals above \(\Omega\). This allows one to iteratively take limits and fixed points, and therefore is significantly stronger than DoRI. == Main system == The main system is divided into infinitely many subsystems. The zeroth subsystem has limit [[Epsilon numbers|\(\varepsilon_0\)]], the first subsystem has limit [[Bachmann-Howard ordinal|BHO]]<nowiki>, and the second subsystem's limit is greater than the limit of DoR. While an obsolete analysis suggested the limit was beyond a nonprojectible ordinal, the current best lower bound on its strength is slightly greater than an ordinal \(\alpha\) which is \(\beta\)-stable, where \(\beta > \alpha\) and \(\beta\) is \(\omega_{\beta^+}^{\mathrm{CK}}\)-stable.</nowiki> == Sources == dcb28312d8c564126e5dad059da7db71c34eb60d 0 143 780 619 2024-09-24T06:45:04Z C7X 9 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (\(L\) does not have the [[covering property]]).<ref>Any text about Jensen's covering theorem</ref> * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref> * There is a proper class of nontrivial elementary embeddings \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\).<ref name=":0">W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref> * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\).<ref name=":0" /> While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable (stable for first-order formulae?) - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. Alternatively, \(0^\sharp\) may be defined as a sound mouse (iterable premouse), or as an Ehrenfeucht-Mostowski blueprint. There is an analysis of sharps in the terminology of Giard's flowers.<ref>J. P. Aguilera, "[https://eprints.whiterose.ac.uk/178745/6/BoundingSharps_alt.pdf Boundedness theorems for flowers and sharps]". Proceedings of the American Mathematical Society vol. 150 (2022)</ref> 1429bf552f9613220db980b356b3bb23eac930f1 781 780 2024-09-24T06:45:18Z C7X 9 wikitext text/x-wiki Zero sharp is a [[sharp]] for the constructible universe [[Constructible hierarchy|\(L\)]], which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every uncountable cardinal \(\kappa\): * \(|I \cap \kappa| = \kappa\) * For any finite sequences \(\alpha_1 < \alpha_2 < \cdots < \alpha_n\) and \(\beta_1 < \beta_2 < \cdots < \beta_n\) of elements of \(I \cap \kappa\) and any formula \(\varphi\) in the language of set theory with \(n\) free variables, \(L_\kappa\) satisfies \(\varphi(\alpha_1, \alpha_2, \cdots, \alpha_n)\) iff it satisfies \(\varphi(\beta_1, \beta_2, \cdots, \beta_n)\) * For every \(a \in L_\kappa\), there is a formula \(\varphi\) in the language of set theory and \(\alpha_1, \alpha_2, \cdots, \alpha_n \in I \cap \kappa\) so that, for all \(x \in L_\kappa\), \(x = a\) iff \(L_\kappa\) satisfies \(\varphi(x, \alpha_1, \alpha_2, \cdots, \alpha_n)\). Clause 2 is abbreviated as "\(I \cap \kappa\) is a set of indiscernibles for \(L_\kappa\)", and clause 3 is abbreviated "every \(a \in L_\kappa\) is definable in \(L_\kappa\) with parameters from \(I \cap \kappa\)". Such a set \(I\) is called a class of Silver indiscernibles. It is known that the following are equivalent: * \(0^\sharp\) exists. * There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (\(L\) does not have the [[covering property]]).<ref>Any text about Jensen's covering theorem</ref> * For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\). * Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]]. * \(\aleph_\omega^V\) is regular in \(L\). * There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref> * There is a proper class of nontrivial elementary embeddings \(j: L \to L\), all with different critical points. * For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\). * Every uncountable cardinal is inaccessible in \(L\).<ref name=":0">W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref> * There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\).<ref name=":0" /> While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: * \(\aleph_\omega\) is totally stable (stable for first-order formulae?) - in fact, every uncountable cardinal is totally stable. * If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). * There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. And "\(0^\sharp\) exists" is strictly implied by the following: * Chang's conjecture * There is a singular strong limit cardinal \(\kappa\) so that \(2^\kappa > \kappa^+\). * There is a [[weakly compact cardinal]] \(\kappa\) so that \((\kappa^+)^L < \kappa\). * There is a Ramsey cardinal. If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles. Alternatively, \(0^\sharp\) may be defined as a sound mouse (iterable premouse), or as an Ehrenfeucht-Mostowski blueprint. There is an analysis of sharps in the terminology of Girard's flowers.<ref>J. P. Aguilera, "[https://eprints.whiterose.ac.uk/178745/6/BoundingSharps_alt.pdf Boundedness theorems for flowers and sharps]". Proceedings of the American Mathematical Society vol. 150 (2022)</ref> f6c394a0cba921f4489c987098b691a87399f94e 2 243 782 2024-11-01T12:11:50Z Unexian1296 41 Start my page. Will edit soon with SiSS wikitext text/x-wiki Hello, hello, hello! I am Unexian1296! I believe myself to know quite a bit, as I can expand both Bashicu Matrix System and Y-Sequence. I'm quite proud of myself for that. I also have my own sequence system, which can get up to \(\varepsilon_\omega\), interestingly enough. I have some other stuff, too, but they'd be better to rest on my [https://googology.miraheze.org/wiki/User:Unexian1296 googology wiki page] 99c0c31a82c15faf5f6ca3b2577a20dc22f76d2f 0 56 783 603 2024-11-02T06:47:04Z OfficialURL 10 fix typo wikitext text/x-wiki The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend [[Cantor normal form]] by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the [[epsilon numbers]]. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \). Analogously to [[Cantor normal form]], every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary VNF is known as \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions. Ordinals unreachable from below via binary Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to [[Epsilon numbers|\( \alpha \mapsto \varepsilon_\alpha \)]] or [[Epsilon numbers|\( \alpha \mapsto \zeta_\alpha \)]]. Ordinals beyond \( \Gamma_0 \) can be written using a variadic extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the [[Small Veblen ordinal]], and can be extended, through transfinite amounts of entries (formalized via finitely-supported [[Ordinal function|functions on ordinals]]), to the [[Large Veblen ordinal]], and through "rows" and "planes" of entries to reach the [[Bachmann-Howard ordinal]], in which one allows taking fixed points of amounts of entries.<ref>https://arxiv.org/abs/2310.12832v1?</ref> However, for ordinals beyond the Large Veblen ordinal, [[Ordinal collapsing function|ordinal collapsing functions]] are typically considered more efficient.<ref>Rathjen https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, pp.10--11</ref> == References == 16cd52b929227801cdf5c9a15980c593e2516346 0 51 784 513 2024-11-02T08:52:03Z OfficialURL 10 wikitext text/x-wiki In set theory, the '''ordinal numbers''' or '''ordinals''' are an extension of the [[natural numbers]] that describe the order types of [[Well-ordered set|well-ordered sets]]. A set \( S \) is '''well-ordered''' if each non-empty \( T \subseteq S \) has a least element. For example, any finite set of real numbers is well-ordered, as well as the set of natural numbers, while neither the set of nonnegative rationals nor the set of integers is well-ordered. This is in contrast to the [[Cardinal|cardinals]], which only describe cardinality, and which are applicable to non-well-ordered sets. The class of ordinal numbers is denoted by \(\mathrm{Ord}\) or \(\mathrm{On}\). The idea of ordinals as a transfinite extension of the counting numbers was first invented by Georg Cantor in the 19th century. ==Von Neumann definition== In a pure set theory such as [[ZFC]], we need a way to define ordinals as objects of study. The Von Neumann definition of ordinals does this, by associating each ordinal \( \alpha \) is defined as the set of all ordinals less than \( \alpha \). In particular: * \(0 := \{\}\) * \(\alpha+1 := \alpha \cup \{\alpha\}\) * If \(X\) is a set of ordinals, then \(\bigcup X\) By associating the natural number \(0\) with the ordinal \(0\), \(1\) with \(0+1 = \{0\}\), \(2\) with \(0+1+1 = \{0,1\}\), and so on, the natural numbers can be embedded inside the ordinals. However, the set of natural numbers (which is its own union) is also an ordinal, and commonly written as \(\omega\). One convenient property of this definition of ordinals is that \(\alpha < \beta\) can be easily defined to mean \(\alpha \in \beta\), and thus \(\omega\) is an ordinal bigger than all the natural numbers. By continuing on this way, we can form a never-ending ladder of ordinals, and assign an order type to any well-ordered set. One is able to define ordinals without this recursive definition. In particular, in ZFC, the two following statements are equivalent to \(\alpha\) being an ordinal. * \(\alpha\) is a transitive [[set]] and all elements of \(\alpha\) are transitive. * \(\alpha\) is a transitive set and the \(\in\) relation restricted to \(\alpha\) is a strict well-order. Using [[Supertask|supertasks]], it is possible to count up to any given countable ordinal in a finite amount of time. For example, to count to \(\omega\), one takes 30 seconds to count from 0 to 1, 15 seconds to count from 1 to 2, 7.5 seconds to count from 2 to 3, 3.75 seconds to count from 3 to 4, and so on. After a minute, all numbers \(< \omega\) will be exhausted. In general, one can count to \(\omega \cdots n\) in \(n\) seconds. By iterating this process another layer, one can count to \(\omega^2\): one takes 1 minute to count from 0 to \(\omega\), 30 seconds to count from \(\omega\) to \(\omega 2\) (so 15 seconds to count from \(\omega\) to \(\omega + 1\), 7.5 seconds to count from \(\omega + 1\) to \(\omega + 2\), and so on), then 15 seconds to count from \(\omega 2\) to \(\omega 3\), and so on. In general, any countable ordinal can be counted to in a finite amount of time. However, it is impossible to count to [[Countability|\(\omega_1\)]] in any finite amount of time. This idea is closely related to [[Matchstick diagram|matchstick diagrams]] - it is possible to draw a diagram for an arbitrary countable ordinal but not \(\omega_1\). The order type of a well-ordered set is intuitively its "length". Alternatively, one may think of the order type of a set \(X\) as the smallest ordinal so that the ordinals below are sufficient to number the elements of \(X\), while preserving the order. For example, the order type of a singleton if 1, since pairing 0 with the single element of that set is a [[bijection]] and preserves order, therefore, the ordinals below 1 are sufficient to order-preservingly number the elements of the singleton. For finite sets, the order type and cardinality are equal. In particular, the order type of (the von Neumann representation) any natural number \(n\) is defined as \(n\). In general, any ordinal is its own order type. But also many non-ordinal objects have order types. For example, say we were to reorder the natural numbers by putting all the even numbers first, followed by the odd numbers. This is still well-ordered, and has order type \(\bigcup\{\omega+n: n < \omega\}\), also written \(\omega \cdot 2\). == Ordinal arithmetic == We can do arithmetic with ordinals like so: * \(\alpha + 0 = \alpha\) * \(\alpha + (\beta + 1) = (\alpha + \beta) + 1\) * If \(\beta\) is not \(0\) or a successor to another ordinal (in which case it is called a limit ordinal), \(\alpha + \beta = \bigcup\{\alpha+\gamma: \gamma < \beta\}\) One can see that this agrees with the usual definition of arithmetic for the natural numbers when \(\alpha\) and \(\beta\) are finite. Similarly: * \(\alpha \cdot 0 = 0\) * \(\alpha \cdot (\beta + 1) = (\alpha \cdot \beta) + \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha \cdot \beta = \bigcup\{\alpha \cdot \gamma: \gamma < \beta\}\) Again, this agrees with the usual definition. Lastly: * \(\alpha^0 = 1\) * \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\) * If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\) Ordinal arithmetic is well-defined by the axioms of union, pairing and replacement. There are helpful visual representations for these, namely with [[Matchstick diagram|matchstick diagrams]]. For example, \(\alpha + \beta\) can be visualized as (a diagram for) \(\alpha\), followed by a copy of (a diagram for) \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines, and they therefore have not only the same [[Cardinal|cardinality]] but the same order type. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over [[ZFC]]: "if \(X\) and \(Y\) are well-ordered sets with order types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order type \(\alpha + \beta\)". For ordinal multiplication, \(\alpha \cdot \beta\) can be imagined as \(\beta\), with each individual line in \(\beta\) replaced with a copy of \(\alpha\). For example, \(\omega \cdot 2\), is two lines, with each individual line replaced with a copy of \(\omega\), i.e: 2 copies of \(\omega\), or \(\omega + \omega\). \(\alpha^\beta\) may be described in terms of functions \(f:\beta\to\alpha\) with finite support.<ref>J. G. Rosenstein, ''Linear Orderings'' (1982). Academic Press, Inc.</ref> Note that, generally, if one of \(\alpha\) and \(\beta\) is infinite, then \(\alpha + \beta\) will have the same cardinality as \(\max(\alpha, \beta)\), but, as we mentioned, not necessarily the same order type. A similar result holds for ordinal multiplication and addition, and it can be shown by "interlacing" well-ordered sets with the respective order types. An exercise is to formally find a [[bijection]] from \(\max(\alpha, \beta)\) to \(\alpha + \beta\), assuming the former is [[infinite]]. == Equivalence class definition == Another common definition of ordinals, often used in settings not based on set theory, is as equivalence classes of well-orders. Namely, we say that two well-ordered sets are order-isomorphic (iso- for "same" and morphic for "form" or "shape") if there is a way of relabelling the elements of the first set into elements of the second set, so that the order is preserved. Note that this implies the two sets have the same size, but is a strictly stronger notion: the video linked in the previous section shows that \(\omega\) and \(\omega + 1\) have the same size, yet aren't order-isomorphic. Order-isomorphism is used to give the definition of order type: the order type of \(X\) is the unique ordinal \(\alpha\) which it is order-isomorphic to. However, outside of this context, it is used to give an alternate, simpler (yet formally more troublesome) definition of ordinals. Namely, an ordinal can be defined as the equivalence class of sets under order-isomorphism. For example, \(\omega\) is defined as the class of all sets which are order-isomorphic to the natural numbers. The issue is that all ordinals, other than zero, are now proper classes, which makes formal treatment more difficult. 8f999ad98428528d9a73d6310d2e0f37d7069b4e 0 17 788 642 2024-11-05T02:25:18Z CreeperBomb 30 wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|\(\alpha\)-recursion theory]] (the study of generalising recursion on the natural numbers to on \(L_\alpha\) for [[admissible]] ordinals \(\alpha\)) * [[B-recursion theory|\(\beta\)-recursion theory]] (the generalisation of \(\alpha\)-recursion theory to non-admissible \(\alpha\)) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is the extended Buchholz function unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \), PTO of EFA and ID0 + Exp *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]] * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'', edited by R. Schindler, Ontos Series in Mathematical Logic (2010, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies \(\mathsf{AQI}\), arithmetical quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" />{{verification failed}} * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ 5ec120095f9ed93604f8508a82bbc1bfe2de2df3 789 788 2024-11-05T06:05:15Z CreeperBomb 30 Described omega^omega wikitext text/x-wiki Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties. As such, apeirology is linked to: * [[Set theory]] (which includes study of large cardinals) * [[A-recursion theory|\(\alpha\)-recursion theory]] (the study of generalising recursion on the natural numbers to on \(L_\alpha\) for [[admissible]] ordinals \(\alpha\)) * [[B-recursion theory|\(\beta\)-recursion theory]] (the generalisation of \(\alpha\)-recursion theory to non-admissible \(\alpha\)) * [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded) * Googology (which translates recursive ordinal notations into systems for constructing large finite numbers). Below we list some milestone ordinals. == Countable ordinals == In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> The \(\psi\) is the extended Buchholz function unless specified. * [[0]], the smallest ordinal * [[1]], the first successor ordinal * [[omega|\( \omega \)]], the first limit ordinal * [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal * \( \omega^{3} \), PTO of EFA and ID0 + Exp *<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.--> * [[omega^omega|\( \omega^{\omega} \)]], the second infinite multiplicative principal ordinal and PTO of RCA₀ and WKL₀ * [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA₀^{(sort out page)} * [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]^{(decide if own page)} * [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal * [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref> * [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal^{(decide if keep)} * [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) * [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) * [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) * [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension * [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction * [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal * [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion * \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below) * \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM * \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF) * \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L * <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals * PTO of \( \Pi^1_2 \)-comprehension * PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset * PTO of \( \text{KP} + "\omega_1 \) exists \( " \) * PTO of \( \text{ZFC} \) * [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal * \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π¹₂ Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>^p.24 * The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref>^(p.3) * The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" />^(p.3) * The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref>^(p.13) * The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> * The least \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\) * The least \( (\cdot 2) \)-stable ordinal * The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" />^(p.4) * The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky--> * The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" />^(pp.3,9)<ref name=":0" /> * The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" />^(p.20) * The least \( (^++1) \)-stable ordinal<ref name="OrderOfReflection" />^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)} * The least (next recursively inaccessible ordinal)-stable ordinal * The least (next recursively Mahlo ordinal)-stable ordinal * The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \) * The least doubly \( (+1) \)-stable ordinal<ref name=":0" />^(p.4) * The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" />^p.24 * The least nonprojectible ordinal<ref name=":0" />^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref>^(p.218) * The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" />^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" />^(p.221) * HIGHER STABILITY STUFF GOES HERE^{(sort out)} * Some Welch stuff here * [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals ** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal ** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \) ** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal ** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability * The least ordinal in \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'', edited by R. Schindler, Ontos Series in Mathematical Logic (2010, p.338).</ref> * The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" /> * The least admissible \(\alpha\) so that \(L_\alpha\) satisfies \(\mathsf{AQI}\), arithmetical quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" />{{verification failed}} * Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" /> * The smallest [[gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref> *<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. --> * Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> * Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> * Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".<ref name=":0" />^(p.6) <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> <!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?--> * Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" />^(p.6) * <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer--> * Least height of model of ZFC<ref name=":0" />^(p.6) <!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>--> * Least stable ordinal<ref name=":0" />^(p.6)<ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref>^(p.9), this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" />^p.23 * Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" /> * The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" />^(p.8) The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref>^{Corollary 21} <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19--> == Uncountable ordinals == * [[omega_1|\( \Omega \)]], the smallest [[uncountable]] ordinal Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist: * \( I \), the smallest [[inaccessible cardinal]] * \( M \), the smallest [[Mahlo cardinal]] * \( K \), the smallest [[weakly compact cardinal]] == References == __NOEDITSECTION__ d23888d95ead0cea16e135ad648f1221cce5c4b8 0 1 790 751 2024-11-07T01:22:44Z CreeperBomb 30 Linked to Googology Wiki wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta and Other Links === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' * [https://googology.miraheze.org/wiki/Main_Page Googology Wiki] </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> b403aaa8178e0654fdaeb1a18230e2c3c1bfdb53 795 790 2024-11-08T16:57:48Z RhubarbJayde 25 Undo revision [[Special:Diff/790|790]] by [[Special:Contributions/CreeperBomb|CreeperBomb]] ([[User talk:CreeperBomb|talk]]) wikitext text/x-wiki <!-- making sure the wiki doesnt get inactivated --> <!-- Main Layout Div --> <div style="display:flex;flex-direction:column;align-items:center;"> <big>Welcome to the '''Apeirology Wiki''', a wiki dedicated to the infinite.</big> <br> <!-- Link Lists Div --> <div style="width:72%;display:flex;flex-direction:row;"> <!-- Important Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Important Links === </div> * [[List of functions]] * [[List of ordinals]] </div> <!-- New Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === New to Apeirology? === </div> * '''[https://neugierde.github.io/cantors-attic Cantor's Attic]''' </div> <!-- Other Links Div --> <div style="flex-grow:1;"> <div style="text-align:center;"> === Meta === </div> * Join our '''[https://discord.gg/r3F2WAQEth Discord!]''' </div> </div> </div> __NOTOC__ <!-- Remove the table of contents --> __NOEDITSECTION__ <!-- Remove the section edit links --> e2b43ff20f800c007ae08aa90b03c1ad7f9c7348 0 50 792 654 2024-11-07T01:37:57Z CreeperBomb 30 Explained a bit better wikitext text/x-wiki {{stub}} '''Cantor normal form''' is a standard form of writing ordinals. Specifically, ordinals are written as \( \omega^{\beta_1} + \omega^{\beta_2} + \dots + \omega^{\beta_k} \), where \( \beta_1 \ge \beta_2 \ge \dots \ge \beta_k \) and \( k \ge 0 \) is an integer. Cantor's normal form theorem states that every ordinal \( \alpha \) can be written uniquely in this form. When \( \alpha \) is smaller than [[Epsilon numbers|\( \varepsilon_0 \)]], the exponents \( \beta_1 \) through \( \beta_k \) are all strictly smaller than \( \alpha \). Thus, Cantor normal form can be iterated to form an [[ordinal notation system]] for ordinals less than \( \varepsilon_0 \). 9d516566c0fbf765a83ed5a817af81fd64f9f226 0 223 793 724 2024-11-07T06:12:05Z CreeperBomb 30 Major stub wikitext text/x-wiki {{stub}} A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>[https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> b8de7cbfbca763d1930fc2225b581a8337567ae2 797 793 2024-11-10T21:39:06Z C7X 9 wikitext text/x-wiki {{stub}} A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>[https://logicdavid.github.io/files/mthesis.pdf#page=21]</ref> ==Sources== {{reflist}} 68bafc5b9e19fc8f844d02d01f69da87cf65cc74 798 797 2024-11-10T21:42:05Z C7X 9 wikitext text/x-wiki {{stub}} A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.{{citation needed}} A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\). <ref>D. Schrittesser, "[https://logicdavid.github.io/files/mthesis.pdf#page=21 \(\Sigma^1_3\)-Absoluteness in Forcing Extensions]" (2004).</ref> ==Sources== <references /> 7cafb898e94295fc5bcd0f2f2e82950d3d214a49 0 166 794 726 2024-11-07T21:22:50Z CreeperBomb 30 stub marker wikitext text/x-wiki {{stub}} The axiom of infinity is a common mathematical axiom included in theories such as [[Kripke-Platek set theory]] or [[ZFC]]. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that [[Omega|\(\omega\)]] exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define [[Ordinal|ordinals]]. For example, \(V_\omega\), the set of [[Hereditarily finite set|hereditarily finite sets]], is a model of [[ZFC]] minus the axiom of infinity. d7fc9a6f254df854836340385c165aafd2af67c5 0 74 799 291 2024-11-12T00:15:53Z CreeperBomb 30 /* Von Neumann ordinals */ wikitext text/x-wiki The '''natural numbers''', or '''counting numbers''', are a system of numbers which includes the positive integers \( 1, 2, 3, \dots \), and under some definitions also includes [[zero]]. If zero is to be considered a natural number, which is usually the case in set theory, the natural numbers are precisely the finite [[ordinal]]s. ==Encodings== The natural numbers are fundamental objects in mathematics, and thus different areas of math have different conventions of encoding or defining them. In many areas such as geometry or number theory, they are taken as primitives and not formally treated ===Von Neumann ordinals=== In [[ZFC]] and other set theories without urelements, we can define natural numbers by applying the definition of Von Neumann ordinals to finite ordinals. In this view, each natural number is the set of previous naturals: \( 0 = \varnothing \) and \( n + 1 = \{0, \dots, n\} \). Under this definition, the cardinality of a natural number precisely equals the number. ===Zermelo ordinals=== [[:wikipedia:Ernst Zermelo|Ernst Zermelo]] provided an alternative construction of the natural numbers, encoding \( 0 = \varnothing \) and \( n + 1 = \{ n \} \) for \( n \ge 0 \). Unlike the Von Neumann ordinals, Zermelo's encoding can only be used to represent finite ordinals. ===Frege and Russell=== During the early development of foundational philosophy and logicism, [[:wikipedia:Gottlob Frege|Gottlob Frege]] and [[:wikipedia:Bertrand Russell|Bertrand Russell]] proposed defining a natural number \( n \) as the equivalence [[class]] of all sets with [[cardinality]] \( n \). This definition cannot be realized in ZFC, because the classes involved are [[proper class]]es, except for \( n = 0 \). ===Church numerals=== In the [[lambda calculus]], the standard way to encode natural numbers is as Church numerals, developed by [[:wikipedia:Alonzo Church|Alonzo Church]]. In this encoding, each natural number \( n \) is identified with a function that returns the composition of its input with itself \( n \) times: \( 0 := \lambda f. \lambda x. x \), \( 1 := \lambda f. \lambda x. f x \), \( 2 := \lambda f. \lambda x. f (f x) \), etc. ==Theories of arithmetic== Axiomatic systems that describe properties of the naturals are called arithmetics. Two of the most popular are [[Peano arithmetic]] and [[second-order arithmetic]]. ==Algebraic properties of the natural numbers== The natural numbers (including zero) are closed under addition and multiplication. They satisfy commutativity and associativity of both operations, and distributivity of multiplication over addition. They form a monoid under addition, which is the free monoid with one generator. In addition, positive naturals form a monoid under multiplication -- the free monoid with countably infinite generators, which are the prime numbers. 5724151a9d76dc6c2be3eec7b4cf922171d7cfee 0 174 800 725 2024-12-05T08:35:05Z C7X 9 I'm not sure there are no nonstandard models of ACA0 - Simpson's book has a chapter on "non-omega-models" and it mentions ACA0 wikitext text/x-wiki Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> The second-order extension of Peano arithmetic is [[second-order arithmetic]], a significantly more expressive system. One subsystem, \(\mathrm{ACA}_0\) (arithmetical comprehension axiom) is first-order conservative over Peano arithmetic, meaning a first-order sentence is provable by Peano arithmetic iff it is provable by \(\mathrm{ACA}_0\).<ref name="Simpson09">S. G. Simpson, ''Subsystems of Second-Order Arithmetic'' (2009).</ref><supCorollary IX.1.6</sup> Both Peano arithmetic and \(\mathrm{ACA}_0\) have non-\(\omega\)-models,<ref>For Peano arithmetic this is a common exercise, provable using compactness. For \(\mathrm{ACA}_0\), take a nonstandard model \(M\) of Peano arithmetic, and use lemma IX.1.3 of Simpson's ''Subsystems of Second-Order Arithmetic'' (2009) to construct a model of \(\mathrm{ACA}_0\) with the same (nonstandard) first-order part as \(M\).</ref> meaning models whose first-order parts are not isomorphic to the standard natural numbers. The set of finite von Neumann [[Ordinal|ordinals]], paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto a \cup \{a\}\) is a model of Peano arithmetic, and one of the most "natural" models of Peano arithmetic. Alternatively, the set of Zermelo ordinals, paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto \{a\}\) is a model of Peano arithmetic. However, the natural functions and relations \(+\), \(\cdot\) and \(<\) in this structure are more complex to describe. Peano arithmetic, minus the axiom schema of induction and plus the axiom \(\forall y (y = 0 \lor \exists x (S(x) = y))\) (which is a theorem of Peano arithmetic but requires induction), is known as Robinson arithmetic, and has proof-theoretic ordinal [[Omega|\(\omega\)]]. As mentioned previously, the axiom schema of induction gives Peano arithmetic a majority of its strength, which is shown by the fact that it has proof-theoretic ordinal [[Epsilon numbers|\(\varepsilon_0\)]], famously shown by Gentzen. Similarly, Robinson arithmetic is unable to show the function \(f_\omega\) in the fast-growing hierarchy is total, while the least rank of the fast-growing hierarchy which outgrows all computable functions provably total in Peano arithmetic is \(f_{\varepsilon_0}\). 1dd5d6657ab4eccc0ec98cb91fe0dda6b9da5389 801 800 2024-12-05T08:35:45Z C7X 9 wikitext text/x-wiki Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> The second-order extension of Peano arithmetic is [[second-order arithmetic]], a significantly more expressive system. One subsystem, \(\mathrm{ACA}_0\) (arithmetical comprehension axiom) is first-order conservative over Peano arithmetic, meaning a first-order sentence is provable by Peano arithmetic iff it is provable by \(\mathrm{ACA}_0\).<ref name="Simpson09">S. G. Simpson, ''Subsystems of Second-Order Arithmetic'' (2009).</ref>^{Corollary IX.1.6} Both Peano arithmetic and \(\mathrm{ACA}_0\) have non-\(\omega\)-models,<ref>For Peano arithmetic this is a common exercise, provable using compactness. For \(\mathrm{ACA}_0\), take a nonstandard model \(M\) of Peano arithmetic, and use lemma IX.1.3 of Simpson's ''Subsystems of Second-Order Arithmetic'' (2009) to construct a model of \(\mathrm{ACA}_0\) with the same (nonstandard) first-order part as \(M\).</ref> meaning models whose first-order parts are not isomorphic to the standard natural numbers. The set of finite von Neumann [[Ordinal|ordinals]], paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto a \cup \{a\}\) is a model of Peano arithmetic, and one of the most "natural" models of Peano arithmetic. Alternatively, the set of Zermelo ordinals, paired with [[Empty set|\(\emptyset\)]] and \(a \mapsto \{a\}\) is a model of Peano arithmetic. However, the natural functions and relations \(+\), \(\cdot\) and \(<\) in this structure are more complex to describe. Peano arithmetic, minus the axiom schema of induction and plus the axiom \(\forall y (y = 0 \lor \exists x (S(x) = y))\) (which is a theorem of Peano arithmetic but requires induction), is known as Robinson arithmetic, and has proof-theoretic ordinal [[Omega|\(\omega\)]]. As mentioned previously, the axiom schema of induction gives Peano arithmetic a majority of its strength, which is shown by the fact that it has proof-theoretic ordinal [[Epsilon numbers|\(\varepsilon_0\)]], famously shown by Gentzen. Similarly, Robinson arithmetic is unable to show the function \(f_\omega\) in the fast-growing hierarchy is total, while the least rank of the fast-growing hierarchy which outgrows all computable functions provably total in Peano arithmetic is \(f_{\varepsilon_0}\). 871198cf8cb52cbde865944fb9454f7a59e04638 0 115 802 336 2024-12-05T08:38:58Z C7X 9 Remove induction wikitext text/x-wiki Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail of ordinal analysis, and many believe it can be done using [[Bashicu matrix system|BMS]]. ==Reverse mathematics== One of the primary interests regarding \(Z_2\) is the study of its subsystems, rather than the whole. This is part of a program called reverse mathematics. Since rational numbers, real numbers, complex numbers, continuous functions on the reals, countable groups, and more can be defined in the language of second-order arithmetic, it turns out many classical theorems in number theory, real analysis, topology, abstract algebra and group theory are provable in \(Z_2\), and most even in weak subsystems! The "big five" are the following:<ref>Subsystems of Second Order Arithmetic, Simpson, S.G., ''Perspectives in Logic'', 2009, ''Cambridge University Press''</ref> * \(\mathrm{RCA}_0\): recursive comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^0_1\)-formulae and induction restricted to \(\Sigma^0_1\)-formulae. \(\mathrm{RCA}_0\) has proof-theoretic ordinal [[Omega^omega|\(\omega^\omega\)]], and it can prove the following famous results: the Baire category theorem, the intermediate value theorem, the soundness theorem, the existence of an algebraic closure of a countable field, the existence of a unique real closure of a countable ordered field. * \(\mathrm{WKL}_0\): weak König's lemma, i.e. \(\mathrm{RCA}_0\) with the additional axiom "every infinite binary tree has an infinite branch" \(\mathrm{WKL}_0\) has the same proof-theoretic ordinal as \(\mathrm{RCA}_0\), but is able to prove some non-induction related theorems which \(\mathrm{RCA}_0\) can't, such as: the Heine/Borel covering lemma, every continuous real-valued function on [0, 1], or even any compact metric space, is bounded, the local existence theorem for solutions of (finite systems of) ordinary differential equations, Gödel’s completeness theorem, every countable commutative ring has a prime ideal and Brouwer’s fixed point theorem. * \(\mathrm{ACA}_0\): arithmetical comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^1_0\)-formulae and induction restricted to the first-order case \(\mathrm{ACA}_0\) and Peano arithmetic have the same first-order consequences and thus the same proof-theoretic ordinal: namely, [[Epsilon numbers|\(\varepsilon_0\)]]. Not much has been said regarding \(\mathrm{ACA}_0\)'s ordinary, non-number-theoretical consequences. * \(\mathrm{ATR}_0\): arithmetical transfinite recursion, i.e. \(\mathrm{ACA}_0\) with the additional axiom "every arithmetical operator can be iterated along any countable well-ordering" and induction restricted to the first-order case \(\mathrm{ATR}_0\) has the proof-theoretic ordinal [[Feferman-Schütte ordinal|\(\Gamma_0\)]], which is part of the reason why the ordinal in question is claimed to be the limit of what can be predicatively defined. \(\mathrm{ATR}_0\) can prove the following: any two countable well orderings are comparable, any two countable reduced Abelian p-groups which have the same Ulm invariants are isomorphic, and that every uncountable closed, or analytic, set has a perfect subset. * \(\Pi^1_1 \mathrm{-CA}_0\): \(\Pi^1_1\)-comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Pi^1_1\)-formulae and induction restricted to the first-order case \(\Pi^1_1 \mathrm{-CA}_0\) has a significantly higher proof-theoretic ordinal than the previous entries - namely, [[Buchholz ordinal|\(\psi_0(\Omega_\omega)\)]]. It can prove the following: every countable Abelian group is the direct sum of a divisible group and a reduced group, the Cantor/Bendixson theorem, a set is Borel iff it and its complement are analytic, any two disjoint analytic sets can be separated by a Borel set, coanalytic uniformization, and more. There are also even stronger systems such as \(\Pi^1_1 \mathrm{-TR}_0\), which is \(\Pi^1_1 \mathrm{-CA}_0\) with the axiom "every \(\Pi^1_1\)-definable operator can be iterated along any countable well-ordering", \(\Pi^1_2 \mathrm{-CA}_0\), and more. The former has proof-theoretic ordinal [[Extended Buchholz ordinal|EBO]], while the latter's proof-theoretic ordinal hasn't been precisely calibrated but has been bound.<ref>Determinacy and \(\Pi^1_1\) transfinite recursion along \(\omega\), Takako Nemoto, 2011</ref><ref>An ordinal analysis of \(\Pi_1\)-Collection, Toshiyasu Arai, 2023</ref> 524b2fee5f13b299e97274056d9c863e85a54696 803 802 2024-12-05T08:41:22Z C7X 9 heeheeheehoohoohoo wikitext text/x-wiki Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail of ordinal analysis, and many believe it can be done using [[Bashicu matrix system|BMS]]. An ordinal analysis of \(Z_2\) was claimed by Arai in 2023<ref>T. Arai, "[https://arxiv.org/abs/2311.12459 An ordinal analysis of \(\Pi_N\)-Collection]" (2023).</ref> and Towsner by 2024.<ref>H. Towsner, "[https://arxiv.org/abs/2403.17922 Proofs that Modify Proofs]" (2024).</ref> ==Reverse mathematics== One of the primary interests regarding \(Z_2\) is the study of its subsystems, rather than the whole. This is part of a program called reverse mathematics. Since rational numbers, real numbers, complex numbers, continuous functions on the reals, countable groups, and more can be defined in the language of second-order arithmetic, it turns out many classical theorems in number theory, real analysis, topology, abstract algebra and group theory are provable in \(Z_2\), and most even in weak subsystems! The "big five" are the following:<ref>Subsystems of Second Order Arithmetic, Simpson, S.G., ''Perspectives in Logic'', 2009, ''Cambridge University Press''</ref> * \(\mathrm{RCA}_0\): recursive comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^0_1\)-formulae and induction restricted to \(\Sigma^0_1\)-formulae. \(\mathrm{RCA}_0\) has proof-theoretic ordinal [[Omega^omega|\(\omega^\omega\)]], and it can prove the following famous results: the Baire category theorem, the intermediate value theorem, the soundness theorem, the existence of an algebraic closure of a countable field, the existence of a unique real closure of a countable ordered field. * \(\mathrm{WKL}_0\): weak König's lemma, i.e. \(\mathrm{RCA}_0\) with the additional axiom "every infinite binary tree has an infinite branch" \(\mathrm{WKL}_0\) has the same proof-theoretic ordinal as \(\mathrm{RCA}_0\), but is able to prove some non-induction related theorems which \(\mathrm{RCA}_0\) can't, such as: the Heine/Borel covering lemma, every continuous real-valued function on [0, 1], or even any compact metric space, is bounded, the local existence theorem for solutions of (finite systems of) ordinary differential equations, Gödel’s completeness theorem, every countable commutative ring has a prime ideal and Brouwer’s fixed point theorem. * \(\mathrm{ACA}_0\): arithmetical comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Delta^1_0\)-formulae and induction restricted to the first-order case \(\mathrm{ACA}_0\) and Peano arithmetic have the same first-order consequences and thus the same proof-theoretic ordinal: namely, [[Epsilon numbers|\(\varepsilon_0\)]]. Not much has been said regarding \(\mathrm{ACA}_0\)'s ordinary, non-number-theoretical consequences. * \(\mathrm{ATR}_0\): arithmetical transfinite recursion, i.e. \(\mathrm{ACA}_0\) with the additional axiom "every arithmetical operator can be iterated along any countable well-ordering" and induction restricted to the first-order case \(\mathrm{ATR}_0\) has the proof-theoretic ordinal [[Feferman-Schütte ordinal|\(\Gamma_0\)]], which is part of the reason why the ordinal in question is claimed to be the limit of what can be predicatively defined. \(\mathrm{ATR}_0\) can prove the following: any two countable well orderings are comparable, any two countable reduced Abelian p-groups which have the same Ulm invariants are isomorphic, and that every uncountable closed, or analytic, set has a perfect subset. * \(\Pi^1_1 \mathrm{-CA}_0\): \(\Pi^1_1\)-comprehension axiom, i.e. \(Z_2\) with comprehension restricted to \(\Pi^1_1\)-formulae and induction restricted to the first-order case \(\Pi^1_1 \mathrm{-CA}_0\) has a significantly higher proof-theoretic ordinal than the previous entries - namely, [[Buchholz ordinal|\(\psi_0(\Omega_\omega)\)]]. It can prove the following: every countable Abelian group is the direct sum of a divisible group and a reduced group, the Cantor/Bendixson theorem, a set is Borel iff it and its complement are analytic, any two disjoint analytic sets can be separated by a Borel set, coanalytic uniformization, and more. There are also even stronger systems such as \(\Pi^1_1 \mathrm{-TR}_0\), which is \(\Pi^1_1 \mathrm{-CA}_0\) with the axiom "every \(\Pi^1_1\)-definable operator can be iterated along any countable well-ordering", \(\Pi^1_2 \mathrm{-CA}_0\), and more. The former has proof-theoretic ordinal [[Extended Buchholz ordinal|EBO]], while the latter's proof-theoretic ordinal hasn't been precisely calibrated but has been bound.<ref>Determinacy and \(\Pi^1_1\) transfinite recursion along \(\omega\), Takako Nemoto, 2011</ref><ref>An ordinal analysis of \(\Pi_1\)-Collection, Toshiyasu Arai, 2023</ref> bd05b92edff240c0c0a52d0c02c5541bb56fca60 0 112 804 592 2025-01-14T02:45:53Z IDoNotExist 16 wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \varepsilon_0 \), \( \theta(\Omega 2) = \zeta_0 \), \( \theta(\Omega^2) = \Gamma_0 \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions (extensions are possible, but unformalized, and it gets very difficult once \(\Omega_2\) is introduced.) Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \varphi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. Bachmann's method was extended to use higher cardinals, e.g. to use \(\Omega_n\) for all finite \(n\) by Pfeiffer in 1964 and to use \(\Omega_\alpha\) for \(\alpha<I\) by Isles in 1970,<ref>Buchholz, Feferman, Pohlers, Sieg, ''Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies''. Lecture Notes in Mathematics (1981). Springer Berlin Heidelberg, ISBN 9783540386490.</ref> but with similarly cumbersome definitions.<ref name="RathjenArt" />^p.11 A modern "recast", proposed by Michael Rathjen<ref name="RathjenArt">Rathjen, Michael. "[https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]".</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref>^{Maybe not? Look at things around p.11 more}<!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal 4a07466437ab062bc61e5ef05817c9ab5aa48bca 805 804 2025-01-14T02:48:44Z IDoNotExist 16 fixed errors wikitext text/x-wiki An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as [[Uncountable|\( \Omega \)]] or [[Church-Kleene ordinal|<nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki>]] to smaller, recursive ordinals such as the [[Small Veblen ordinal|SVO]]. The primary idea is that, at the point of [[epsilon numbers]] and beyond, especially at the level of [[Strongly critical ordinal|strongly critical ordinals]], representation systems for ordinals require complex arrays and processes of fixed point-taking. Instead, one imbues a certain level of impredicativity into the definition and uses large ordinals that would normally be entirely unreachable from below and instead use them as "diagonalizers", with higher levels taking fixed points of lower levels. The simplest application of this idea is Chris Bird's \( \theta \) function, which uses polynomials in \( \Omega \) to encode finitary Veblen arrays. For example, we have \( \theta(\Omega) = \Gamma_0 \), \( \theta(\Omega 2) = \varphi(2,0,0) \), \( \theta(\Omega^2) = \varphi(1,0,0,0) \), and so on. However, this isn't entirely formal, and its limit is the [[small Veblen ordinal]], small comparatively to other ordinal collapsing functions (extensions are possible, but unformalized, and it gets very difficult once \(\Omega_2\) is introduced.) Most ordinal collapsing functions, both in the literature and apeirological circles, are instead defined using a construction which is typically referred to as a \( C \)-set construction or Skolem hull construction of all the ordinals that can be "predicatively" built up using available operations, with some restrictions, and then letting the collapse of an ordinal \( \alpha \) be the limit of those ordinals. == History == The first ordinal collapsing function in the literature was Bachmann's \( \varphi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. Bachmann's method was extended to use higher cardinals, e.g. to use \(\Omega_n\) for all finite \(n\) by Pfeiffer in 1964 and to use \(\Omega_\alpha\) for \(\alpha<I\) by Isles in 1970,<ref>Buchholz, Feferman, Pohlers, Sieg, ''Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies''. Lecture Notes in Mathematics (1981). Springer Berlin Heidelberg, ISBN 9783540386490.</ref> but with similarly cumbersome definitions.<ref name="RathjenArt" />^p.11 A modern "recast", proposed by Michael Rathjen<ref name="RathjenArt">Rathjen, Michael. "[https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]".</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that the countable ordinals constructible from \(\Omega\) and the set of ordinals below \( \max(1, \rho) \) using the following operations are all less than \(\rho\): addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition. * Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). * For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+\delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) * \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). * \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). The limit of this notation is the Bachmann-Howard ordinal, \( \psi_\Omega(\varepsilon_{\Omega+1}) \). This definition has been relativized to more powerful ordinal collapsing functions, most using large cardinal axioms to "diagonalize" over lower levels, such as [[Buchholz's psi-functions]], which collapse \( \aleph_\nu \) for \( \nu \leq \omega \), and even up to Michael Rathjen's most recent OCF, which collapses an analogue of a cardinal \( \kappa \) which is \( \delta \)-shrewd, where \( \delta \) is the next weakly inaccessible cardinal after \( \kappa \). == Remarks == Rathjen has remarked upon a small detail of various ordinal collapsing functions, which he calls pictorial collapse. In particular, he says that, in the OCF defined in the "History" section, \( \psi_\Omega(\alpha) \) can be viewed as the "\( \alpha \)th collapse of \( \Omega \)" since, as he puts it, the order-type of \( \Omega \) as viewed within \( C^\Omega(\alpha, \psi_\Omega(\alpha)) \) is actually \( \psi_\Omega(\alpha) \).<ref>M. Rathjen, Proof Theory (Stanford encyclopedia of Philosophy), special case of definition 5.11</ref> The same property applies to [[Buchholz's psi-functions]]<ref>W. Buchholz, ''A New System of Proof-Theoretic Ordinal Functions'', Annals of Pure and Applied Logic, vol. 32, pp.195-207, (1986)</ref>, Rathjen's OCF collapsing a [[Mahlo cardinal|weakly Mahlo cardinal]]<ref>Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.</ref>, and Rathjen's OCF collapsing a [[weakly compact cardinal]].<ref>Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).</ref> == Use of nonrecursive countable ordinals == It may seem circular to use an OCF whose definition presumes existence of uncountable or large cardinals in the ordinal analysis of a theory much weaker than ZFC. However all instances of uncountable cardinals in the definition of the OCF may be replaced with nonrecursive countable ordinals, at the expense of much difficulty in proving the relevant theorems about the OCF. For example, Rathjen has defined an OCF for analyzing KPM that uses [[Admissible ordinal|admissible ordinals]] in place of regular cardinals, and recursively Mahlo ordinals in place of Mahlo cardinals.<ref>Realm of Ordinal Analysis, near the end</ref> The main technique for proving the relevant theorems about \( \psi_\pi(\alpha) \) (e.g. that \( \psi_\pi(\alpha)<\pi \)) becomes to use a \( \pi \)-recursive coding scheme to code the representable ordinals that are \( >\pi \) into members of \( L_\pi \). This coding must respect the ordering of the \( \psi_\kappa(\beta) \), which itself has not yet been verified.<ref name="Rathjen94">Possible citation? [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf]</ref>^{Maybe not? Look at things around p.11 more}<!--This citation is for previous sentence also--> This is in contrast to the cardinal-based definition of \( \psi \), in which case these can be proven by simple arguments like cardinality arguments.<ref>M. Rathjen, "How to develop proof-theoretic ordinal functions on the basis of admissible ordinals" (1998), MSC-1991 classification 03F13/03F35. Accessed 1 September 2023. Possibly in Mathematical Quarterly vol. 39 (1993)?</ref> In addition, some of the relations recursive definitions are performed along are well-founded but non-set-like.<ref name="Rathjen94" />^p.5 == Quantifier complexity == The quantifier complexity of many OCFs seems to be \( \Sigma_1 \). This holds for Buchholz's and Bachmann's OCFs,<ref>The only citation I have is a Discord message, https://discord.com/channels/206932820206157824/655959490755035169/861665386705059870</ref> and Arai's OCF for \( \Pi_n \)-reflection.<ref>T. Arai, "[https://arxiv.org/abs/1907.07611v1 A simplified ordinal analysis of first-order reflection]", proposition 2.7. arXiv version (2019), accessed 31 August 2023.</ref> The quantifier complexity of Arai's OCF for \( \textsf{KP}\ell^r+\exists M(\textrm{isTrans}(M)\land M\prec_{\Sigma_1}V) \) is \( \Delta_1 \),<ref>T. Arai, "[https://arxiv.org/abs/2208.12944 An ordinal analysis of a single stable ordinal]", proposition 2.11. arXiv version (2023), accessed 31 August 2023.</ref> however the quantifier complexity of Arai's OCF for KP+\( \Pi_1 \)-collection is \( \Delta_1(St) \).<ref>T. Arai, "[https://arxiv.org/abs/2112.09871 An ordinal analysis of \(\Pi_1\)-Collection]", proposition 3.8. arXiv version (2023), accessed 31 August 2023.</ref> == List == Following Bachmann's \( \psi \), there have been various generalizations and strengthenings. These include: * Feferman's \( \theta \)-functions * [[Buchholz's psi-functions|Buchholz's \( \psi \)-functions]], a simplification of Feferman's \( \theta \)-functions * Schütte and Simpson's addition-free versions of Buchholz's functions, denoted \( \pi_i \)<ref name="VanDerMeeren15">J. Van der Meeren, "[https://core.ac.uk/download/pdf/55770155.pdf Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems]". PhD dissertation (2015), accessed 31 August 2023.</ref>^pp.20--23 * Gordeev's \( D_\nu \) functions<ref name="VanDerMeeren15" />^pp.2--3 * Maksudov's extended Buchholz \( \psi \)-functions, an extension of Buchholz's \( \psi \)-functions * Madore's \( \psi \)-function, a further simplification of Buchholz's \( \psi \)-functions * Bird's \( \theta \)-function * Weiermann's \( \vartheta \), a variant of Buchholz's \( \psi \)-functions with nicer behaviour * Wilken's \( \vartheta \), which does not need standard forms,<ref>G. Wilken, "[https://www.sciencedirect.com/science/article/pii/S0168007206001175 Ordinal arithmetic based on Skolem hulling]", p.131. Annals of Pure and Applied Logic vol. 145, iss. 2 (2007), pp.130--161.</ref> & Wilken and Weiermann's \( \bar{\vartheta} \), variants of Weiermann's \( \vartheta \) * Jäger's \( \psi \)-function, an extension of Bachmann's \( \psi \) to the level of [[Inaccessible cardinal|weakly inaccessible cardinals]] * The Jäger-Buchholz function, a simplification of Jäger's \( \psi \)-function * Rathjen's \( \psi \) function, an extension of Jäger's \( \psi \)-function to the level of [[Mahlo cardinal|weakly Mahlo cardinals]] * Rathjen's \( \Psi \) function, an extension of his \( \psi \)-function to the level of [[Weakly compact cardinal|weakly compact cardinals]] * Duchhardt's \( \Psi \) function, an extension of Rathjen's \( \Psi \)-function to the level of \( \Pi^1_2 \)-indescribable cardinals * Stegert's \( \Psi \) functions, extensions of Duchhardt's \( \Psi \)-function to the level of \( \Pi^2_0 \)-indescribable cardinals and analogues of pseudo-\((\cdot 2)\)-stable ordinals * Rathjen's \( \Psi \) functions, extensions of his \( \Psi \)-function to the level of analogues of pseudo-\((\cdot 2)\)-stable ordinals and analogues of \((\alpha^{+I})\)-stable ordinals * Arai's first \( \psi \) function, a simplification of Stegert's first \( \Psi \)-function * Arai's second \( \psi \) function, an extension of his first \( \psi \)-function to the level of analogues of nonprojectible ordinal a59603fe1de991fe7c5054b101a8b1029948f8d5 1 232 806 753 2025-01-24T20:52:56Z C7X 9 /* Flood of new accounts */ new section wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) :id like to see proof of that [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 20:22, 25 March 2024 (UTC) :: https://apeirology.com/wiki/Special:Contributions/Cobsonwabag. Examples include: https://apeirology.com/wiki/Main_Page?oldid=746, https://apeirology.com/wiki/Proving_well-orderedness?oldid=744, https://apeirology.com/wiki/Burali%E2%80%93Forti_paradox?oldid=697, https://apeirology.com/wiki/Buchholz%27s_psi-functions?oldid=696, and almost every other edit for vandalism, and https://apeirology.com/wiki/File:Coinslot.png?oldid=658 for explicit content [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 05:08, 26 March 2024 (UTC) :::ip grabbers [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 12:20, 26 March 2024 (UTC) == Flood of new accounts == See [https://apeirology.com/wiki/Special:RecentChanges?hidebots=1&limit=50&days=7&enhanced=1&urlversion=2 Recent changes] [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:52, 24 January 2025 (UTC) 9e0fe98a40ee020cf023439177385a6e163e5535 807 806 2025-01-24T20:53:14Z C7X 9 /* Flood of new accounts */ wikitext text/x-wiki == User:Cobsonwabag == We need to ban User:Cobsonwabag for vandalizing a ton of pages and adding images of private parts. [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 17:14, 25 March 2024 (UTC) :id like to see proof of that [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 20:22, 25 March 2024 (UTC) :: https://apeirology.com/wiki/Special:Contributions/Cobsonwabag. Examples include: https://apeirology.com/wiki/Main_Page?oldid=746, https://apeirology.com/wiki/Proving_well-orderedness?oldid=744, https://apeirology.com/wiki/Burali%E2%80%93Forti_paradox?oldid=697, https://apeirology.com/wiki/Buchholz%27s_psi-functions?oldid=696, and almost every other edit for vandalism, and https://apeirology.com/wiki/File:Coinslot.png?oldid=658 for explicit content [[User talk:CreeperBomb|CreeperBomb]] ([[User:CreeperBomb|talk]]) 05:08, 26 March 2024 (UTC) :::ip grabbers [[User:Cobsonwabag|Cobsonwabag]] ([[User talk:Cobsonwabag|talk]]) 12:20, 26 March 2024 (UTC) == Flood of new accounts == See [https://apeirology.com/wiki/Special:RecentChanges?hidebots=1&limit=50&days=7&enhanced=1&urlversion=2 Recent changes], should something be done to deal with this? [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:52, 24 January 2025 (UTC) 0da1ff93c04ba83a2524c2e5d4a4a1a21cb05e4e